text:
\begin{document}
\date{
April 12, 2021}

\author{Nasrin Altafi\\[.05in]
{\ns Department of Mathematics, KTH Royal Institute of Technology, S-100 44 Stockholm, Sweden.
}\\[.2in] Anthony Iarrobino\\[.05in]
{\ns Department of Mathematics, Northeastern University, Boston, MA 02115,
 USA.
}\\[.2in]
Leila Khatami\\[0.05in]
{\ns Union College, Schenectady, New York,
 12308, USA.}
 \\[.2in]
Joachim Yam\'{e}ogo\\[0.05in]
{\ns Universit\'e C\^ote d'Azur, CNRS, LJAD, FRANCE.}
}

\title{Jordan types for graded Artinian algebras in height two
\footnote{\textbf{Keywords}: Artinian algebra, Hilbert function, hook code, Jordan type, partition, cellular decomposition, graded ideal. \textbf{2010 Mathematics Subject Classification}: Primary: 13E10;  Secondary: 05A17, 05E40, 13D40, 14C05.
\quad Email addresses: {\sf nasrinar@kth.se, a.iarrobino@northeastern.edu, khatamil@union.edu, joachim.yameogo@unice.fr}}}
\maketitle
\abstract
We let $A=R/I$ be a standard graded Artinian algebra quotient of $R={\sf k}[x,y]$, the polynomial ring in two variables over a field ${\sf k}$ by an ideal $I$, and let $n$ be its vector space dimension.
The Jordan type $P_\ell$ of a linear form $\ell\in A_1$ is the partition of $n$ determining the Jordan block decomposition of the multiplication on $A$ by $\ell$ -- which is nilpotent. The first three authors previously determined which partitions of $n=\dim_{\sf k}A$ may occur as the Jordan type for some linear form $\ell$ on a graded complete intersection Artinian quotient $A=R/(f,g)$ of $R$, and they counted the number of such partitions for each complete intersection Hilbert function $T$ \cite{AIK}.\par
 We here consider the family $\G_T$ of graded Artinian quotients  $A=R/I$ of $R={\sf k}[x,y]$, having arbitrary Hilbert function $H(A)=T$. The cell $\mathbb V(E_P)$ corresponding to a partition $P$ having diagonal lengths $T$ is comprised of all ideals $I$ in $R$ whose initial ideal is the monomial ideal $E_P$ determined by $P$.  These cells give a decomposition of the variety $\G_T$ into affine spaces.  We determine the generic number $\kappa(P)$ of generators for the ideals in each cell $\mathbb V(E_P)$, generalizing a result of \cite{AIK}. In particular, we determine those partitions for which $\kappa(P)=\kappa(T)$, the generic number of generators for an ideal defining an algebra $A$ in $\G_T$. We also count the number of partitions $P$ of diagonal lengths $T$ having a given $\kappa(P)$. A main tool is a combinatorial and geometric result  allowing us to split $T$ and any partition $P$ of diagonal lengths $T$ into simpler $T_i$ and partitions $P_i$, such that $\mathbb V(E_P)$ is the product of the cells $\mathbb V(E_{P_i})$, and $T_i$ is single-block:  $G_{T_i}$ is a Grassmannian.

\tableofcontents\vskip 0.3cm\noindent
\section{Introduction.}
Let $A$ be a standard-graded Artinian algebra $A=\mathop{\oplus}\limits_{i=0}^{\sf j}A_i$ over a field $\F$,  and let $\ell\in A_1$ be a linear form. The multiplication map $m_\ell: A \to A:  a \to \ell\cdot a$ is nilpotent. The \emph{Jordan type} $P_\ell=P_{\ell,A}$ is a partition of $n=\dim_\F A$, giving the Jordan block decomposition of the multiplication map $m_\ell$.
We consider standard graded Artinian algebra quotients $A=R/I$ where $I$ is an ideal of $ R={\sf k}[x,y]$, the polynomial ring $R$ in two variables over $\sf k$.
We will assume that $I_1=0$, so the codimension $\dim_\F A_1$ of $A$ is two. The \emph{order} of the graded ideal $I$ is the lowest degree of a (non-zero) element.
\par
The Hilbert function $T=H(A)$ of such a graded Artinian algebra  $A$ in codimension two is a sequence of the following form
\begin{equation}\label{Teq}
T=\left(1,2,\dots , d, t_d,t_{d+1},\dots ,t_{\sf j},0\right) \text { where } d\ge t_d\ge t_{d+1}\ge \cdots \ge t_{\sf j}>0,
\end{equation}
where $t_i=\dim_{\sf k} A_i$, $\sf j$ is the {\emph{socle degree}} of $T$, $d$ is the \emph{order} of $T$ -- the order of any ideal $I$, so $H(R/I)=T$ -- and $n=|T|=\sum t_i=\dim_{\sf k}A$.\par
\par
The first three authors in \cite{AIK} determined all possible Jordan types $P_\ell$ of linear forms $\ell$ for complete intersection (CI) graded Artinian algebras of height two: they assumed that the field $\sf k$ either has characteristic zero, or is infinite of characteristic $p>{\sf j}$, the socle degree of the algebra. In this paper we make the same assumption on characteristic because we use a standard-basis result due to J. Brian\c{c}on and A. Galligo in showing Lemma \ref{corner-kick-off-lemma}, that requires the restriction.
Our main results in this paper generalize those of \cite{AIK} to all height two Hilbert functions. \par

Let $P$ be a partition and let $E_P$ be the monomial ideal determined by the Ferrers diagram of $P$. The \emph{diagonal lengths} of the Ferrers diagram of $P$ is just the Hilbert function $T_P=H(R/E_P)$ (Definition \ref{EPdef}).  The set $\mathcal P(T)$ of partitions having diagonal lengths $T$ -- or, equivalently -- the set of monomial ideals $E_P$ such that $H(R/E_P)=T$, has been studied by the second and last author in \cite{IY}, as well as by others, including \cite{Ev,Con}. Let $\mathbb V(E_P)$ be the affine cell that parametrizes the ideals of $R$ having initial ideal $E_P$ (Definition \ref{celldef}). Recall that the family $\G_T$ of graded Artinian quotients $A=R/I$, where $I$ is an ideal of $R={\sf k}[x,y]$, for which the Hilbert function $H(A)=T$ is a smooth projective variety $\G_T$, that is locally an affine space of known dimension (Proposition \ref{GTdimthm}).
 L.~G\"{o}ttsche showed that the cells $\mathbb V(E_P)$ for $P\in \mathcal P(T)$ form a cellular decomposition of $\G_T$ \cite{Got}; this followed the analogous results of G. Ellingsrud and S.A. Str{\o}mme for the punctual Hilbert scheme of projective space $\mathbb P^2$, using the A. Bialynicki-Birula theorem \cite{ES1,ES2,Bia}. This cellular decomposition was as well studied by G.~Gotzmann \cite{Gm2}, and the fourth author \cite{jy-1,jy-2}. Generators and relations for height two graded ideals have been studied by many, as \cite{Bu,brian, brian-gal,MR,Con}; J.O. Kleppe studies a scheme analogue of $G_T$ \cite{Kl}, L.~Evain discusses an equivariant Hilbert scheme, involving weights of the variables \cite{Ev};  some combinatorics of cells in \cite{IY} are seen in a larger context in~\cite{LW}. The cells $\mathbb V(E_P)$ and their connection with generators and relations of ideals have also been studied by A.~Conca and G.~Valla in  \cite{CoVa}.\vskip 0.2cm\par\noindent
 By a \emph{generic} element of an irreducible algebraic variety $X$ we will mean an element belonging to a certain non-empty Zariski-dense open subset $U\subset X$.\par
The main results of this paper, {\bf Theorems \ref{kappathm} and \ref{componentTheorem}}, give $\kappa(P)$, the minimum number of generators for the ideal $I$ defining a generic element $A=R/I$ in the cell $\mathbb{V}(E_P)$, for each partition $P\in \mathcal P(T)$, where $T$ is a Hilbert function of a height two graded Artinian algebra, namely a sequence of the form given in Equation~\eqref{Teq}. In {\bf Theorem
\ref{multicountthm}}, we give the number of partitions $P\in \mathcal P(T)$ with $\kappa(P)=k$, for any positive integer $k$.

\vskip 0.2cm\par\noindent
{\bf Summary.}
{\it Decomposition of cells.},
 We first prove in Section \ref{prodsec} a  new combinatorial and geometric result allowing us to associate to an arbitrary Hilbert function $T$ and to any partition $P$ of diagonal lengths $T$ their \emph{components}, a set of simpler, single-block sequences $T_i$ and partitions $P_i$ of diagonal lengths $T_i$. We show that the affine cell $\mathbb V(E_P)$ of $\G_T$ is in a natural way the product of its single-block components, the cells $\mathbb V(E_{P_i})$ of $\G_{T_i}$  ({\bf Theorem~\ref{projectionthm}}).  This decomposition is closely related to the \emph{hook codes} that had been studied in \cite{IY} and that we define in Section  \ref{hookcodesec}.\vskip 0.2cm\par\noindent
 {\it Single block case.}
In the \emph{single-block} case the Hilbert function sequence $T$ of Equation \eqref{Teq} satisfies $d={\sf j}$, so
 \begin{equation}\label{single-block1eq}
 T=(1,2,\ldots,d,t,0),
 \end{equation}
 that is $t_i=i+1$ for $0\le i\le d-1$ and $ t_d=t$.
 An ideal $I\subset R$ defining an algebra $A=R/I$ of Hilbert function $T$ of  Equation~\eqref{single-block1eq} satisfies
  \begin{equation}\label{single-block2eq}
  I=V\oplus \mathfrak m^{d+1},
  \end{equation}
  where we let $V=I_d\subset R_d$ and $\mathfrak m$ is the maximal ideal of~$R$.
 Thus, the projective variety $G_T$ in the single-block case is isomorphic to the Grassmannian $\Grass(s,R_d), s=d+1-t$, parametrizing $s$-dimensional subspaces $V\subset R_d$.  In {\bf Theorem \ref{kappathm}} we determine the integer $\kappa(P)$, the number of generators of a generic ideal in the cell $\mathbb V(E_P)$, for single-block partitions, and in {\bf Theorem~\ref{countingthm}}, we give the number of \emph{special} single-block partitions $P$, namely partitions $P$ with $\kappa(P)>\kappa(T)$, the generic - and minimum - number of generators for an ideal defining an algebra $A$ in $\G_T$. More generally, in {\bf Corollary \ref{singlecountcor}} we determine the number of partitions $P\in \mathcal P(T)$ with $\kappa(P)=k$, for any positive integer $k$.
\vskip 0.2cm\par\noindent
 {\it General case.}
 In {\bf Theorem \ref{componentTheorem}} we determine $\kappa(P)$ for an arbitrary partition $P\in \mathcal P(T)$ in terms of the $\kappa(P_i)$ of their single-block components. In {\bf Theorem~\ref{multicountthm}} we give the number of partitions $P\in \mathcal P(T)$ with $\kappa(P)=k$, for any positive integer $k$. We show that $P$ is special ($\kappa(P)\not= \kappa(T)$) if and only if some component $P_i$ is special ({\bf Theorem \ref{componentthm}}). We determine $\kappa(P)$ for arbitrary Jordan types $P$ ({\bf Theorem \ref{multicountthm}}); this result also determines the number of special partitions in $\mathcal P(T)$ (Corollary \ref{countspecialcor}), and we recover the number of CI partitions shown in \cite{AIK} (Corollary \ref{CIJordantypecor}).

\vskip 0.2cm\par\noindent
{\it Hook code}. We explain in Section \ref{hookcodesec} the \emph{hook code} for partitions $P$ of diagonal lengths $T$.  The hook code of $P$ is a sequence $\mathfrak Q(P)=\left(\mathfrak h_d(P),\ldots  ,\mathfrak h_{\sf j}(P)\right)$ of partitions-in-a-box $\mathfrak {\mathfrak B}_i(T), d\le i\le \sf j$, where the box $\mathfrak {\mathfrak B}_i(T)=(\delta_{i+1})\times (1+\delta_i)$: that is, the Ferrers diagram of each $\mathfrak h_i(P)$ has at most $\delta_{i+1}$ rows and $(1+\delta_i)$ columns (Definition \ref{hookcodedef}). The partitions of diagonal lengths  $T$ are completely determined by their hook code $\mathfrak Q(P)$. Also, the component partition $P_i$ of diagonal lengths $T_i$ has as its hook code the degree-$i$ component  $\mathfrak h_i(P)$ of $\mathfrak Q(P)$ (Lemma \ref{hookP(i)lem}).
\vskip 0.2cm\noindent
  \par
The proofs involve a careful study of standard generators and relations for the ideals $I$ defining algebras in the cell $\mathbb V(E_P)$, using in particular the hook code of a partition $P$. We then compare these invariants to those for the partition $P:x$ corresponding to the ideals $I:x$.  This allows us to compare $\kappa(P)$ with $\kappa (P:x)$, and we thus determine how to compute $\kappa(P)$ from the hook code $\mathfrak Q(P)$.
\section{Cells of the variety $G_T$ and their hook codes.}\label{GTchapter}
\subsection{The variety $\G_T$ and the cells $\mathbb V(E_P)$.}\label{GTsec}
We need some basic notions from \cite{IY,IY2} (see also \cite[\S 4.1]{AIK}).\par

Recall that we consider graded Artinian quotients $A=R/I$, where $I$ is an ideal of $R={\sf k}[x,y]$ the polynomial ring over an arbitrary field $\sf k$. The Hilbert function of $A$ is the sequence $H(A)=(1,t_1,\ldots, t_{\sf j} )$ where $t_i=\dim_{\sf k} A_i$ and $\sf j$ is the socle degree of $A$ that is $ A_{\sf j}\not=0, A_{{\sf j} +1}=0$. The family of all such quotients having Hilbert function $H(A)=T$ is denoted by $\G_T$, which has a natural structure of subvariety $\G_T\subset \Pi_{d\le i\le{\sf j}} \Grass(t_i,R_i)$, where $\Grass(t_i,R_i)$ parametrizes quotients $A_i=R_i/I_i$ of vector space dimension $t_i$. Thus we have
$$\iota: \G_T \to \Pi_{d\le i\le {\sf j}} \Grass(t_i,R_i):  A=R/I \to (I_d,I_{d+1},\ldots, I_{\sf j}).$$
We now explain the affine cell decomposition $\G_T=\bigcup_{P\in \mathcal P(T)}\mathbb V(E_P)$ where $P$ runs through the set $\mathcal P(T)$ of partitions having diagonal lengths $T$.  (Theorem \ref{GTdecompthm}).\par
\begin{definition}[The monomial ideal $E_P$ and diagonal lengths of $P$]\label{EPdef}
Given a partition $P=(p_1,p_2,\ldots, p_t)$ of $n=\sum p_i$ where $p_1\ge p_2\ge\cdots \ge p_t$, we let $C_P$ be the set of $n$  monomials that fill the Ferrers diagram $F_P$ of $P$ as follows: for $i\in [1,t]$ the, $i$-th row counting from the top of $F_P$ is filled by the monomials $y^{i-1},y^{i-1}x,\ldots ,y^{i-1}x^{p_i-1}$. We let $E_P$ be the complementary set of monomials to $C_P$ and denote by $(E_P)$ the monomial ideal generated by $E_P$.
 The \emph{diagonal lengths} $T_P$ of $P$ is the Hilbert function $T_P=H(R/E_P)$.\par
 \end{definition}
   In a Ferrers diagram of monomials associated to a partition $P$ of $n$, the $x$-degrees of monomials increase as we go from left to right and the $y$-degrees increase as we go from top to bottom. We count the columns from left to right and the rows from top to bottom.  See Figure \ref{Ferrersfig} for the Ferrers diagram of the partition $P=(5,3,1)$ of diagonal lengths $T=(1,2,3,2,1)$; and Example \ref{flatHFexample} and Figure \ref{5.8figure} for that of $P=\left(10^2,4,3,2^5\right)$.
  \begin{figure}[!h]
 \begin{center}
 \begin{ytableau}
 1&x&x^2&x^3&x^4\cr
 y&yx&yx^2\cr
y^2\cr
 \end{ytableau}
 \caption{Ferrers diagram for $P=(5,3,1)$.}\label{Ferrersfig}
 \end{center}
 \end{figure}\vskip 0.2cm
 \begin{remark}[Sequence of diagonals of $P$]
 Let $P=(p_1,p_2,\ldots, p_t)$ ($p_1\ge p_2\ge\cdots \ge p_t$) be a partition of $n$. For any non-negative integer $i$, let
 $$D_i=\left\{ u \text { such that } 1\leq u \leq i+1\text{ and }  i+2-u\leq p_u \right\}.$$ The sequence $\left(|D_0|, |D_1|,\ldots ,|D_i|,\ldots \right)$ is the diagonal lengths $T_P$.
 \end{remark}
\begin{definition}\label{hookdef}
A \emph{hook} of a partition $P$ is a subset of the Ferrers diagram $F_P$ consisting of a corner monomial $c$, an \emph{arm}
 $(c,xc,\ldots ,\nu=x^{u-1}c)$ and a \emph{leg} $(c,yc,\ldots , \mu= y^{v-1}c)$, such that $x\nu\in E_P$ and $y\mu\in E_P$
 (Figure \ref{hook fig}). The \emph{arm length} is $u$ and the \emph{leg length} is $v$; the hook has arm-leg \emph{difference}
 $u-v$. We term the monomial $\nu$ the \emph{hand}, and the monomial $\mu$ the \emph{foot} of the hook.\par
  \end{definition}
 \begin{figure}[!ht]
\begin{center}
$\begin{array}{|c|c|c|c|}
\hline
c&\phantom{c}&\phantom{c}&\cellcolor{gray1}h\\
\hline
\cline{1-1} \\
\cline{1-1}
\cellcolor{gray1}f\\
\cline{1-1}
\end{array}$
\caption{Difference-one hook with hand $h$, foot $f$, corner $c$, $P=(4,1,1)$.}\label{hook fig}
\end{center}
\end{figure}

\begin{example}\label{431ex} Let $P=(4,3,1)$. $P$ has diagonal lengths $T_P=(1,2,3,2)$. The hook with corner $x$ in the Ferrers diagram $C_P$  has arm length 3, foot length 2, hand $x^3$, foot $yx$, so has (arm $-$ leg)  difference one (Figure \ref{hook1fig}).
\end{example}

\begin{figure}[!ht]
\begin{center}
$\begin{array}{|c|c|c|c|}
\hline
1&\cellcolor{red}x&\cellcolor{gray2}x^2&\cellcolor{gray2}x^3\\
\hline
y&\cellcolor{gray2}yx&yx^2\\
\cline{1-3}
y^2\\
\cline{1-1}
\end{array}$
	\caption{Difference-one hook with corner $x$ for $P=(4,3,1)$.}\label{hook1fig}
\end{center}
\end{figure}\noindent
\begin{definition}[Initial ideal of $I$, and the cell $\mathbb V(E_P)$]\label{celldef}
 The \emph{initial monomial} $\mu(f)=$ in$(f)$ of a form $f=\sum_k a_ky^kx^{i-k}, a_k\in \sf k$ in the $y$-direction is the monomial $\mu(f)=y^sx^{i-s}$ of highest $y$-degree $s$ among those with non-zero coefficients $a_k$.  Given an ideal $I\subset R={\sf k}[x,y]$, defining the Artinian quotient $A=R/I$ we denote by in$(I)$ the ideal
 \begin{equation*}
{\mathrm{in}}(I)= (\{ \mathrm{in}(f), f\in I\})
 \end{equation*}
 generated by the initial monomials of all elements of $I$. We may identify $\mathrm{in}(I)$ with an ideal $E_P$ for a  partition $P=P(I)$ of diagonal lengths $T=H(A)=H(R/E_P)$. \par
 We denote by $\mathbb V(E_P)$ the affine variety parametrizing all ideals $I\subset R$ having initial ideal $E_P$.
 \end{definition}

For the first of the next two results see \cite[Theorems 2.9, 2.12]{Ia}, or \cite[\S 3-B,Theorem 3.12, \S 3-F]{IY}; the cellular decomposition in the second was shown by
 L. G\"{o}ttsche \cite{Got}, and the hook count for the dimension is \cite[Theorem~3.12]{IY}. Further results involving the intersections of closures of cells was shown by the last author in \cite{jy-1,jy-2}, relying in part on methods of J. Brian\c{c}on \cite{brian}. Recall that we denote by $\mathcal P(T)$ the set of all partitions of $n=|T|$ having diagonal lengths $T$.  We denote by $\delta_i(T)$ the difference $\delta_i(T)=t_{i-1}-t_{i}$, for $i\ge d$.
\begin{proposition}[The smooth projective variety $\G_T$]\label{GTdimthm}\cite[Thm. 3.13]{Ia}.
The variety $\G_T$ parametrizing all ideals $I$ of $R={\sf k}[x,y]$ satisfying $H(R/I)=T$ is a smooth irreducible projective variety, that is locally an affine space of dimension $\sum_{i\ge d} (\delta_i+1)(\delta_{i+1})$: it has a connected cover by opens in the same affine space.
\end{proposition}
\begin{theorem}[Cellular decomposition of $\G_T$]\label{GTdecompthm}\cite{jy-1,jy-2}.
The cell $\mathbb V(E_P)$ is an affine space of dimension the total number of difference one hooks in $C_P$, viewed as the Ferrers diagram of the partition $P$ (Definition \ref{EPdef}).\par
The variety $\G_T$ has a finite decomposition into affine cells,
\begin{equation}
\G_T=\bigcup_{P\in \mathcal P(T)}\mathbb V(E_P).
\end{equation}
\end{theorem}
 \subsection{Hook code of $P$.}\label{hookcodesec}
We review the hook code, using results from \cite{IY,IY2}. First, given $T$ satisfying
Equation \eqref{Teq} we define a sequence $\mathfrak B(T)$ of rectangular partitions or boxes. We let $\delta_i(T)=t_{i-1}-t_i$ for $i\ge d(T)$.  We set
\begin{align}\label{BTeq}
\mathfrak B(T)&=\left(\mathfrak B_d(T),\mathfrak B_{d+1}(T),\ldots ,\mathfrak B_{\sf j}(T)\right), \text { where }\notag\\  {\mathfrak B}_i(T)&=(\delta_{i+1})\times (1+\delta_i), \text { a rectangular box},
\end{align}
with height $\delta_{i+1}$ and base $1+\delta_i$: so ${\mathfrak B}_i(T)$ has $\delta_{i+1}$ parts, each $1+\delta_i$.
We order the monomials of degree $i$ by $x^i<x^{i-1}y<\cdots< y^i$ (lex order); certain of these monomials are hands of degree-$i$ hooks of $P$, that is end elements of rows of $C_P$ from Definition \ref{EPdef}, and we order these correspondingly.
\begin{definition}\label{hookcodedef} Suppose that the partition $P$ has diagonal lengths $T$. The (difference-one) \emph{hook code} of $P$ is the sequence
\begin{equation}\label{hookcodeeq}
\mathfrak Q(P)=\left(\mathfrak h_d(P),\ldots  ,\mathfrak h_{\sf j}(P)\right)
\end{equation}
 where $ \mathfrak h_i(P)$ is a partition that enumerates the difference-one hooks of hand-degree $i$, according to their $\delta_{i+1}$ degree-$i$ hands. That is, the $k$-th part of $\mathfrak h_i(P)$ is the number of difference-one hooks having the $k$-th possible degree-$i$ hand. It is not hard to see that the number of difference-one hooks per hand is in the interval $[0,\delta_{i}+1]$, and that is non-increasing: so $\mathfrak h_i(P)$ is a partition, and  $\mathfrak h_i(P)\subset \mathfrak B_i(T)$: the degree-$i$ hook partition fits into the box  $\mathfrak B_i(T)$.\par
Thus, the code is determined by arranging the difference-one hooks of $P$ first, according to their hand-degree $i$, then according to their ``hand monomial,'' determining for each degree $i\in [d,{\sf j}]$ a partition $\mathfrak h_i(P)$.\par
 We denote by $\mathcal Q(T)$ the set of all $({\sf j}+1-d)$-tuples of partitions $(\mathfrak h_d,\ldots, \mathfrak h_{\sf j})$ satisfying, $\mathfrak h_i\subset \mathfrak {\mathfrak B}_i(T)$.
Here $\mathcal Q(T)$ is a lattice under the product structure given by inclusion for each component $\mathfrak h_i$: that is, $\mathcal Q\le \mathcal Q^\prime$ if each $\mathfrak h_i\subset  \mathfrak h_i^\prime$, in the sense that the Ferrers diagram for $\mathfrak h_i$ fits inside that of $  \mathfrak h_i^\prime$.
\end{definition}
\begin{example}[Hook code for $P=(6,3,3,3)$]\label{6333ex}
Let $T=(1,2,3,4,3,2,0)$  where $d=4$; we have  $\delta_4=4-3=1,\delta_5=3-2=1,\delta_6=2-0=2.$ Then $\mathfrak B(T)=\left(\mathfrak B_4,\mathfrak B_5\right)=\left((1\times 2)_4,(2\times 2)_5\right)$. The partition $P=T^\vee=(6,5,3,1)$ has the maximum hook code $\mathfrak Q(P)=\mathfrak B(T)$.
But $P=(6,3^3)$ has the hook code $\mathfrak Q(P)=\left(1_4,(2,1)_5\right)$: the degree four hand monomial is $y^2x^2$ with a single difference-one hook, with corner $y^2$; the degree-5 hand monomials are $x^5$ with two hooks with corners $x,x^4,$ and $y^2x^3$ with one hook, corner $y^3x$. See Figure \ref{6333fig} where we visualize the hooks by showing their corners, blue for degree 4 and red for degree~5.
\end{example}
\begin{figure}
 \begin{center}
$ \begin{array}{|c|c|c|c|c|c|}
\hline
&\cellcolor{red}\bullet&&\phantom{ab}&\cellcolor{red}\bullet&h_{5}\\
\hline
&&\\
\cline{1-3}
\cellcolor{blue}\bullet&&h_4\\
\cline{1-3}
&\cellcolor{red}\bullet&h_5\\
\cline{1-3}
\end{array}$
 \caption{Hook code for $P=(6,3,3,3): \mathfrak Q(P)=\left(1_4,(2,1)_5\right).$ }\label{6333fig}
 \end{center}
 \end{figure}
The following is stated as part of \cite[Theorem 3.27]{IY}, and shown in \cite[Theorem~1.17]{IY2}. Recall that $\mathcal P(T)$ is the set of partitions having diagonal lengths $T$. We denote by $\mathfrak q: \mathcal P(T)\to \mathcal Q(T)$ the hook code map taking $P$ to $\mathfrak Q(P)$. For a partition $\mathfrak h_i\subset \mathfrak B_i$ we denote by $\mathfrak h_i^c$ the complement of $\mathfrak h_i$ in ${\mathfrak B}_i$. For an element $\mathfrak h=(\mathfrak h_d,\ldots ,\mathfrak h_{\sf j})\in \mathcal Q(T)$, we denote by $\mathfrak h^c=(\mathfrak h_d^c,\ldots ,\mathfrak h_{\sf j}^c)$ the complement in $\mathcal B(T)$. Recall that, given $P\in \mathcal P(T)$ we denote by $P^\vee$ the conjugate partition (switch rows and columns in the Ferrers graph of $P$); evidently $P^\vee\in \mathcal P(T)$.
\begin{theorem}\label{PTtoQTthm} Let $T$ satisfy Equation \eqref{Teq}, and let $P\in \mathcal P(T)$. Then in the hook code map $\mathfrak q$, the
$i$-th hook code partition $\mathfrak h_i(P)\subset \mathfrak {\mathfrak B}_i(T)$. Furthermore, the map $\mathfrak q: \mathcal P(T)\to \mathcal Q(T)$ is an isomorphism of sets satisfying $\mathfrak q(P^\vee)=(\mathfrak q(P))^c$.
\end{theorem}
We endow $\mathcal P(T)$ with the structure of a lattice via the isomorphism $\mathfrak q$ to $\mathcal Q(T)$
(see Definition \ref{hookcodedef}). \footnote{There is an alternative poset structure  $\mathcal P_{alt}(T)$ on $\mathcal P(T)$, related to the sequences of degree-$i$ monomials in $C_P$. The inverse $\mathcal Q(T)\cong P(T)\to \mathcal P_{alt}(T)$ is an inclusion of posets, not an isomorphism of lattices as stated incorrectly in \cite[Theorem 3.27]{IY}. See the discussion in \cite{IY2}.}\par
The second and last author showed that the dimension of the cell $\mathbb V(E)$ is the total number of difference-one hooks in the partition $P_E$ determined by $E$ (Theorem \ref{GTdecompthm}): this is just the height of $\mathfrak Q(P)$ in the lattice $\mathcal Q(T)$.  It follows from the A. Bialynicki-Birula result \cite{Bia}  that over the complexes, the Betti numbers of $\G_T$ may be deduced from the cellular decomposition \cite[Theorem~3.28, Theorem~3.29]{IY}. \par
\subsection{The cell $\mathbb V(E_P)$, and its component decomposition.}\label{prodsec}
Throughout this section $T=(1, 2, \dots , d, t_d, \dots , t_{\sf j}, 0)$ will be a Hilbert function satisfying Equation \eqref{Teq} and $P$ will be a partition of diagonal lengths $T$.
We denote by $E_P$ the monomial ideal associated to $P$ and let ${\mathbb V}(E_P)$ be the cell of $G_T$ associated to $P$.
\subsubsection{The component partitions of $P$.}

Let $P$ be a partition with Hilbert function $T=\left(1,\dots ,d,t_d,\dots ,t_{\sf j},0\right)$ and difference-one hook code
 $\mathfrak{Q}(P)=\big( \mathfrak{h}_d,\mathfrak{h}_{d+1},\dots ,\mathfrak{h}_{\sf j}\big)$ (Definition~\ref{hookcodedef}). For $i=d, \dots, {\sf j}$ we let $T_i$ be the single-block Hilbert function
  \begin{align}\label{HFdecomposition}
T_i&=(1, \dots, d_i-1,d_i,t_{d_i},0),
\end{align}
 where $d_i=\delta_i(T)+\delta_{i+1}(T)$ and $\delta_i(T)=t_{i-1}-t_i$.  We
set $t_{d-1}:=d$ and $t_{{\sf j} +1}:=0$.
(There is a shift in degrees, $T_i$ parametrizes ideals of order (initial degree) $d_i=t_{i-1}-t_{i+1}=\delta_i(T)+\delta_{i+1}(T)$).\par
We will next define the $i$-th block of $P$, denoted by $P_i$ (Definition \ref{Pidef1}), and we will show that it is the partition with diagonal lengths $T_i$ and hook code
$\mathfrak{Q}(P_i)=\big(\mathfrak{h}_i\big)=\big(\mathfrak{h}_i(P)\big)$ (Lemma \ref{hookP(i)lem}). A reader on a first look may just use this Lemma as a definition of $P_i$, but for use in proofs, we wish to be more explicit.

Our purpose here is to give a description of ${\mathbb V}(E_P)$ as a product of cells of  ``small'' Grassmannians
(Theorem \ref{projectionthm}).
Recall that for a Hilbert function $T=(1, 2, \dots , d, t_d, \dots , t_{\sf j}, 0)$, we set $\delta_i=t_{i-1}-t_i$ ($d\leq i \leq {\sf j}+1$). \\
Given $P=(p_1, \dots , p_s)$ a partition of diagonal lengths $T$, for each $i$ ($d\leq i\leq {\sf j}$) we construct a vector space $V_i$ of
dimension $n_i=\delta_i+\delta_{i+1}+1$ such that to each element $I$ of the cell ${\mathbb V}(E_P)$ we can associate a subvector space $I_{V_i}$ of $V_i$ with $\displaystyle{\dim_k\left(I_{V_i}\right)=\delta_i+1}$. So the vector space $I_{V_i}$ is an element of
$\Grass(\delta_i+1, V_i)$ and belongs to a cell described by the partition $\mathfrak{h}_i(P)$, the degree-$i$ block of the difference-one hook code of $P$.

\medskip
\noindent
We first define (as in \cite{IY2} but in a more strict way) the
\textsl{horizontal-border monomials} and \textsl{vertical-border monomials} of $E_P$.

\medskip
\noindent
\begin{definition}\label{defi_horiz_vertic_border_monom}
Let $P$ be a partition of diagonal lengths $T$. Denote by $E_P$ the monomial ideal of ${\sf k}[x,y]$ associated to $P$.
\begin{enumerate}[(i).]\item We say a monomial $x^ay^b\in E_P$ is a \textsl{horizontal-border monomial} of $E_P$ if $(b>0$ and $x^ay^{b-1}\notin E_P)$.
\item We say a monomial $x^ay^b\in E_P$ is a \textsl{vertical-border monomial} of $E_P$ if  $a>0$ and $x^{a-1}y^{b}\notin E_P$.
\end{enumerate}

\medskip
\noindent
Denote by $A(P)_i$ the set of degree-$i$ \textsl{horizontal-border monomials} of $E_P$,
$B(P)_i$ the set of degree-$i$ \textsl{vertical-border monomials} of $E_P$ and $(C_P)_i$ the set of degree-$i$ monomials that are not in $E_P$.
\end{definition}
\smallskip
\noindent
\begin{claim}\label{borderclaim}
\begin{enumerate}[(i).]
\item $
\displaystyle{
\left|A(P)_i\right|=
\left\{
\begin{array}{l}
t_{i-1}-t_i+1=\delta_i+1, \ \mbox{if $x^i\in (C_P)_i$}
\\
t_{i-1}-t_i=\delta_i,  \ \mbox{if $x^i\notin (C_P)_i$}
\end{array}
\right.
}
$
\item $
\displaystyle{
\left|B(P)_{i+1}\right|=
\left\{
\begin{array}{l}
t_{i}-t_{i+1}+1=\delta_{i+1}+1, \ \mbox{if $y^{i+1}\in (C_P)_{i+1}$}
\\
t_{i}-t_{i+1}=\delta_{i+1},  \ \mbox{if $y^{i+1}\notin (C_P)_{i+1}$}
\end{array}
\right.
}
$
\end{enumerate}
\end{claim}
\smallskip
\noindent
\begin{proof}[Proof of claim]
(Note: these formulas have been established in \cite{IY2} to define \textsl{difference-$a$ hook partitions}.) \\
One can consider the following maps
\begin{center}
\begin{tabular}{lr}
$\displaystyle{
\begin{array}{llcc}
\varphi_i : & (C_P)_{i-1} & \longrightarrow & (C_P)_i\cup A(P)_i
\\
 & M & \mapsto & yM
\end{array}
} $;
&
$\displaystyle{
\begin{array}{llcc}
\psi_i : & (C_P)_{i} & \longrightarrow & (C_P)_{i+1}\cup B(P)_{i+1}
\\
 & M & \mapsto & xM
\end{array}
} .$
\end{tabular}
\end{center}
The maps $\varphi_i$ and $\psi_i$ are injective. Also note that if $x^i\in (C_P)_i$, than $x^i$ is the only element of
$(C_P)_i\cup A(P)_i $ that is not in the image of $\varphi_i$, so we have
$\left| (C_P)_{i-1}\right|=\left|(C_P)_i \right|+\left|A(P)_i\right|-1$.
If $x^i\notin (C_P)_i$ than $\varphi_i$ is a bijection, thus $\left| (C_P)_{i-1}\right|=\left|(C_P)_i \right|+\left|A(P)_i\right|$.
The formula for $\left|A(P)_i\right|$ follows  from the fact that for any integer $l$, $\left|(C_P)_l \right|=t_l$.
Using the same arguments one can verify the formula for $\left|B(P)_{i+1}\right|$.
\end{proof}
\begin{remark}
\begin{enumerate}[(i).]
\item Each monomial $x^ay^b$ in $A(P)_i$ is just below a degree-$(i-1)$ foot monomial ($x^ay^{b-1}$) of $P$, thus $\left|A(P)_i\right|$ counts the number of degree-$(i-1)$ foot monomials in the Ferrers diagram of $P$.
\item The elements of $B(P)_{i+1}$ are each just right to a degree-$i$ hand monomial, so $\left|B(P)_{i+1}\right|$ counts the number of degree-$i$ hand monomials in the Ferrers diagram of $P$. In Definition \ref{Pidef1} we will consider the first  (numbering from top to bottom-- lex order) $\delta_{i+1}$ degree-$i$ hand monomials of $P$.
\item Recall that $P=(p_1, \dots , p_s)$, with $p_1\geq p_2 \geq \cdots \geq p_s>0$. \\
If $x^i\notin (C_P)_i$, than $p_1\leq i$.
\end{enumerate}
\end{remark}
\begin{definition}[The partition $P_i$ determined by $P$]\label{Pidef1}
For any positive integer $n\in{\mathbb N}$, denote by $\Mon(R_n)$ the set of degree $n$ monomials of $R={\sf k}[x,y]$
and consider the lex order on $\Mon(R_n)$: $x^n<x^{n-1}y<\cdots <xy^{n-1}<y^n$.
 Let $P\in \mathcal P(T)$. If $x^i\in (C_P)_i$ we let $V_{i1}=A(P)_i$. \par
If $x^i\notin (C_P)_i$, we number the $\delta_i$ elements of $A(P)_i$ from top to bottom:
$A(P)_i=\left\{M_{i1}, \dots , M_{i\delta_i} \right\}$. Let $x^ay^b$ be the last (lex order) degree-$i$ \textsl{vertical-border monomial} above  $M_{i1}$. We then let $V_{i1}=A(P)_i\cup\left\{x^ay^b\right\}$ (if $x^i\notin (C_P)_i$).
Thus, $\dim V_{i1}=\delta_i+1$. \par
We now consider the set $V_{i2}$ consisting of the first (lex order) $\delta_{i+1}$ hand monomials in  $(C_P)_i$
and we let $V_i$ be the vector space spanned by $V_{i1}\cup V_{i2}$.

\medskip
\noindent
By definition, $V_i$ has dimension $d_i+1$ where $d_i=\delta_i+\delta_{i+1}$.
So the set $V_{i1}\cup V_{i2}$ is lex ordered and we can consider the one to one correspondence
\begin{equation}\label{monomeq}
s_i : V_{i1}\cup V_{i2} \longrightarrow\Mon(R_{d_i})
\end{equation}
 that respects the lex ordering (the $k$-th element of $V_{i1}\cup V_{i2}$ is associated to the $k$-th element of $\Mon(R_{d_i})$). \par
The vector space $s_i(V_{i1})$ has dimension $\delta_i+1$, so $\langle s_i(V_{i_1}) \rangle+\mathfrak m^{d_i+1}$ is the monomial ideal $(E_{P_i})$ for a unique partition $P_i$ of diagonal lengths the single-block Hilbert function
$T_i=(1, \dots, d_i-1,d_i,t_{d_i},0)$ where $d_i=\delta_i+\delta_{i+1}$ and $t_{d_i}=\delta_{i+1}$.
\end{definition}

\begin{remark}\label{empty_hook_code_block_partition}
For $i\in \left[d,\sf{j} \right]$, it may happen that $t_i=t_{i+1}$, so $\delta_{i+1}=0$. In that case we have:
\begin{enumerate}[(i).]
\item The rectangular box ${\mathfrak B}_i(T)=(\delta_{i+1})\times (1+\delta_i)$ of Equation \ref{BTeq} is empty and so the degree-$i$ partition
$ \mathfrak h_i(P)$ in Equation \ref{hookcodeeq} is empty.
\item $V_{i2}$ is empty and so $s_i(V_{i1})=\Mon(R_{d_i})$.
\item The partition $P_i$  associated to the monomial ideal $\langle s_i(V_{i_1}) \rangle+\mathfrak m^{d_i+1}$ is just the basic triangle
$\Delta_{d_i}=\Delta_{\delta_i}=(\delta_i, \delta_i-1, \ldots , 1)$. Of course, if $\delta_i=0$, then $\Delta_{d_i}=\emptyset$ and
$\langle s_i(V_{i_1}) \rangle+\mathfrak m^{d_i+1}=R$.
\end{enumerate}
\end{remark}
\medskip

The following Lemma follows directly from Definitions \ref{hookcodedef} and \ref{Pidef1}.
\begin{lemma}\label{hookP(i)lem}
The difference-one hook code of $P_i$ is exactly that of the $i$-th component of ${\mathfrak Q}(P)$ (in the difference-one hook code of $P$).
\end{lemma}\noindent
Note, however, the shift in degree:  the difference-one hook code of $P_i$ occurs in the degree $d_i=t_{i-1}-t_{i+1}=\delta_i(T)+\delta_{i+1}(T)$.

\medskip
\noindent
In connection with Lemma \ref{degree_i+1_rel_gen_formulas} where we will be counting the degree $i+1$ relations and corner-monomials of $E_P$ it is interesting to note that,
\begin{lemma}
The bijection $s_i : V_{i1}\cup V_{i2} \longrightarrow \Mon (R_{d_i})$ induces
\begin{enumerate}[(i).]
\item a one to one correspondence between the degree $i+1$ relations of $E_P$ and the degree $d_i+1=\delta_i+\delta_{i+1}+1$
relations of $E_{P_i}$
\item a one to one correspondence between the first  (numbering from top to bottom-- lex order) $\delta_{i+1}$ degree $i+1$ \textsl{vertical-border monomials} of the ideal $E_P$ and the degree $d_i+1$ \textsl{vertical-border monomials} of $E_{P_i}$
\item a one to one correspondence between the degree $i+1$ corner-monomials of $E_P$ and the degree $d_i+1$ corner-monomials of $E_{P_i}$.
\end{enumerate}
\end{lemma}
\begin{proof}
\begin{enumerate}[(i).]
\item Suppose that the monomial $x^{\alpha}y^{i+1-\alpha}$ ($0<\alpha <i+1$) corresponds to a degree $i+1$ relation. Then $x^{\alpha-1}y^{i+1-\alpha}$
is a \textsl{horizontal-border monomial} of $E_P$ and $x^{\alpha}y^{i-\alpha}$ is a \textsl{vertical-border monomial} of $E_P$.
Now, consider the set $(E_P)_{i,\alpha}$ of degree $i$ \textsl{horizontal-border monomials} of $E_P$ that are above $x^{\alpha}y^{i-\alpha}$. If this set is empty, then $x^i\notin (C_P)_i$, so $x^{\alpha}y^{i-\alpha}\in  V_{i1}$ and the degree $i+1$ relation of $E_P$ corresponding to $x^{\alpha}y^{i+1-\alpha}$ is sent to a degree $d_i+1$ relation of $E_{P_i}$. If the set $(E_P)_{i,\alpha}$ is not empty, let $x^{\alpha^{\prime}}y^{i-\alpha^{\prime}}$ be the first element of $(E_P)_{i,\alpha}$ just above $x^{\alpha}y^{i-\alpha}$. By definition, $s_i$ sends $x^{\alpha-1}y^{i+1-\alpha}$ and $x^{\alpha^{\prime}}y^{i-\alpha^{\prime}}$ to two consecutive \textsl{horizontal-border monomials} of $E_{P_i}$, resulting to a degree $d_i+1$ relation of $E_{P_i}$.
\item By Definition \ref{defi_horiz_vertic_border_monom} we know that any degree $i+1$ \textsl{vertical-border monomial} is just to the right of a unique degree $i$ hand monomial. The second statement of the Lemma is then just a remark based on the fact that any element of $V_{i2}$ is a degree $i$ hand  monomial of $P$ and $s_i$ sends the elements of $V_{i2}$ to the degree $d_i$ hand monomials of $E_{P_i}$.
\item Suppose $x^{\alpha}y^{i+1-\alpha}$ is a degree $i+1$ corner monomial of $E_P$.
\begin{enumerate}[a)]
\item If $\alpha=0$, then one can easily see that $y^{d_i}\notin E_{P_i}$, so $y^{d_i+1}$ is a corner-monomial of $E_{P_i}$.
\item If $\alpha>0$ then the corner-monomial $x^{\alpha}y^{i+1-\alpha}$ is also a \textsl{vertical-border monomial} of $E_P$  that will correspond via $s_i$ to a degree $d_i+1$ corner-monomial of $E_{P_i}$.
\end{enumerate}
\end{enumerate}
\end{proof}
\begin{proposition}\label{rm_rows_cols_P_get_P_i}
Using the notation of Definition \ref{Pidef1}, let
$$\displaystyle{V_{i1}\cup V_{i2}=\left\{x^{\alpha_0}y^{\beta_0}, \ldots,  x^{\alpha_{d_i}}y^{\beta_{d_i}}\right\}},$$ where $\alpha_0<\alpha_1<\cdots <\alpha_{d_i}$  (so $\beta_0>\beta_1>\cdots >\beta_{d_i}$). \\
Let $P^{\prime}$ be the partition obtained from $P$ by removing any column of $P$ whose index does not belong to the set $\left\{\alpha_0,\alpha_1,\ldots ,\alpha_{d_i} \right\}$ and any row of $P$ whose index does not  belong to the set $\left\{\beta_0,\beta_1,\ldots ,\beta_{d_i} \right\}$.  Then $P^{\prime}=P_i$.
\end{proposition}
\begin{proof}
In constructing the $i$-th component of ${\mathfrak Q}(P)$ (in the difference-one hook code of $P$), we only need the elements of $V_{i2}$ (degree $i$ hand monomials) and the elements of $V_{i1}$ (related to degree $i$ \textsl{horizontal-border monomials}).The purpose of the bijection $s_i : V_{i1}\cup V_{i2} \longrightarrow \Mon (R_{d_i})$ is to let us focus on these monomials. So by definition of the bijection $s_i$, the partition $P_i$ is obtained from $P$ by ignoring any column of $P$ whose index does not belong to the set $\left\{\alpha_0,\alpha_1,\ldots ,\alpha_{d_i} \right\}$ and any row of $P$ whose index does not  belong to the set $\left\{\beta_0,\beta_1,\ldots ,\beta_{d_i} \right\}$. Deleting these unnecessary rows and columns will result in showing only the relevant degree $i$ hands and degree $i-1$ feet of $P$.
\end{proof}

\begin{remark}\label{comb_PideP_rmk}
Looking at the Ferrers diagram of a partition $P$, we can visualize the set $V_i=V_{i1}\cup V_{i2}$, defined in Definition \ref{Pidef1}, as follows. Consider the diagonal corresponding to degree-$i$ monomials of ${\sf k}[x,y]$, see the grey bubbles in Figure \ref{tobedetermined}. Then the set of degree-$i$ horizontal border monomials of $P$, $A(P)_i$ can correspond to the bubbles outside of the Ferrers diagram that are right below the horizontal edges of $P$, see the blue monomials in Figure \ref{tobedetermined}. If the largest part of $P$ is greater than $i$ (i.e., $x^i\in (C_P)_i$), then $V_{i1}$ is the same as $(A(P))_i$. If the largest part of $P$ is at most $i$ (i.e., $x^i\not \in (C_P)_i$) then $V_{i1}$ also includes the first degree-$i$ vertical border monomial of $P$ that is above all monomials in $(A(P))_i$. This monomial corresponds to a bubble outside of the Ferrers diagram that is immediately to the right of a horizontal edge of $P$, see the red monomial in Figure \ref{tobedetermined}. Finally, monomials in $V_{i2}$ consists of the first $\delta_{i+1}$ hand monomials of $P$. These correspond to bubbles inside the Ferrers diagram that are at the end of a row of $P$, see the blue monomials in Figure \ref{tobedetermined}.

To visualize Proposition \ref{rm_rows_cols_P_get_P_i}, we fill out the degree-$i$ bubbles that correspond to $V_i$ by their monomials. We then remove all rows and columns of $P$ that do not include a filled degree-$i$ bubble, see Figure \ref{tobedetermined}.

\begin{figure}
	\begin{tikzpicture}[scale=.8]
	\draw[very thin] 	(1.5,0)--(7.5,0)
						(1.5,-1)--(7.5,-1)
						(1.5,-2)--(4.5,-2)
						(1.5,-3)--(3.5,-3)
						(1.5,-4)--(3.5,-4)
						(2.5,0)--(2.5,-5)
						(3.5,0)--(3.5,-5)
						(4.5,0)--(4.5,-3)
						(5.5,0)--(5.5,-2)
						(6.5,0)--(6.5,-2)
						(7.5,0)--(7.5,-2);

	\draw[thin,fill=black!5,rounded corners=3mm]
	(1.5,-5) rectangle (2.5,-6)
	(2.5,-4) rectangle (3.5,-5)
	(3.5,-3) rectangle (4.5,-4)
	(4.5,-2) rectangle (5.5,-3)
	(5.5,-1) rectangle (6.5,-2)
	(6.5,0) rectangle (7.5,-1);

\draw[thick, dashed, red]
	                    (1,-1.5)--(8,-1.5)
	                    (6,0.5)--(6, -2.5);

	\draw[black]
	            node at (7,-0.5) {\footnotesize{$x^5$}}
	            node at (3,-4.5) {\footnotesize{$xy^4$}};
	\draw[blue]
			    node at (5,-2.5) {\footnotesize{$x^3y^2$}}
			    node at (4,-3.5) {\footnotesize{$x^2y^3$}}
			    node at (2,-5.5) {\footnotesize{$y^5$}};
	\draw[very thick] (7.5,0)--(7.5,-2)--(4.5,-2)--(4.5,-3)--(3.5,-3)--(3.5,-5)--(1.5,-5)--(1.5,0)--(7.5,0);

	\draw[very thick, red, ->]
		(5,-7)--(5,-10);
	\draw[red]
	    node at (4.5,-7.5) {$P$}
	    node at (4.5,-8.25) {to}
	    node at (4.5,-9.25) {$P_5$};

	\draw[very thin] 	(2,-11)--(7,-11)
						(2,-12)--(7,-12)
						(2,-13)--(5,-13)
						(2,-14)--(4,-14)
						(2,-15)--(4,-15)
						(3,-11)--(3,-15)
						(4,-11)--(4,-15)
						(5,-11)--(5,-13)
						(6,-11)--(6,-12);

	\draw[very thick]
	(2,-11)--(7,-11)--(7,-12)--(5,-12)--(5,-13)--(4,-13)--(4, -15)--(2,-15)--(2,-11);

	\draw[very thin] 	(11.5,0)--(17.5,0)
						(11.5,-1)--(17.5,-1)
						(11.5,-2)--(14.5,-2)
						(11.5,-3)--(13.5,-3)
						(11.5,-4)--(13.5,-4)
						(12.5,0)--(12.5,-5)
						(13.5,0)--(13.5,-5)
						(14.5,0)--(14.5,-3)
						(15.5,0)--(15.5,-2)
						(16.5,0)--(16.5,-2)
						(17.5,0)--(17.5,-2);

	\draw[thin,fill=black!5,rounded corners=3mm]
	(11.5,-6) rectangle (12.5,-7)
	(12.5,-5) rectangle (13.5,-6)
	(13.5,-4) rectangle (14.5,-5)
	(14.5,-3) rectangle (15.5,-4)
	(15.5,-2) rectangle (16.5,-3)
	(16.5,-1) rectangle (17.5,-2)
	(17.5,0) rectangle (18.5,-1);

\draw[thick, dashed, red]
	                    (12,0.5)--(12,-7.5)
	                    (14,0.5)--(14,-5.5)
	                    (15,0.5)--(15,-4.5)
	                    (11,-3.5)--(16, -3.5)
	                    (11,-4.5)--(15, -4.5);

	\draw[red]
	            node at (18,-0.5) {\footnotesize{$x^6$}};
	\draw[black]
	            node at (17,-1.5) {\footnotesize{$x^5y$}};
	\draw[blue]
			    node at (16,-2.5) {\footnotesize{$x^4y^2$}}
			    node at (13,-5.5) {\footnotesize{$xy^5$}};
	\draw[very thick] (17.5,0)--(17.5,-2)--(14.5,-2)--(14.5,-3)--(13.5,-3)--(13.5,-5)--(11.5,-5)--(11.5,0)--(17.5,0);

	\draw[very thick, red, ->]
		(15,-7)--(15,-10);
	\draw[red]
	    node at (15.5,-7.5) {$P$}
	    node at (15.5,-8.25) {to}
	    node at (15.5,-9.25) {$P_6$};

	\draw[very thin] 	(13.5,-11)--(16.5,-11)
						(13.5,-12)--(16.5,-12)
						(13.5,-13)--(14.5,-13)
						(14.5,-11)--(14.5,-14)
						(15.5,-11)--(15.5,-13)
						(16.5,-11)--(16.5,-13);

	\draw[very thick]
	(13.5,-11)--(16.5,-11)--(16.5,-13)--(14.5,-13)--(14.5,-14)--(13.5,-14)--(13.5,-11);

         \end{tikzpicture}
         \caption{Decomposition of partition $P=(6^2, 3, 2^2)$ with diagonal lengths $T=(1,2,3,4,5,3,1)$ and hook code $\mathfrak{Q}(P)=\big((3,1)_5, (2)_6\big)$ into components $P_5=(5,3,2^2)$ and $P_6=(3^2,1)$ of diagonal lengths $T_5=(1,2,3,4,2)$ and $T_6=(1,2,3,1)$, respectively. On the left, the grey bubbles represent degree $5$ monomials, filled with monomials in $V_5$. The monomials in black represent $V_{5,2}$ and the monomials in blue represent $V_{5,1}=(A(P))_5$. On the right, the gray bubbles represent degree $6$ monomials, filled with monomials in $V_6$. The monomials in black represent $V_{6,2}$, the monomials in blue represent $(A(P))_6$ and the monomial in red represents the additional vertical border monomial in $V_{6,1}$. See Remark \ref{comb_PideP_rmk}. }\label{tobedetermined}
\end{figure}

\end{remark}

\begin{example}\label{1stgensEx}
Consider the three-block partition $P=(15, 12^4, 11, 7, 6^2, 5, 3^4)$ with diagonal lengths $T=\left(1,2,\dots ,13,10_{13},6_{14},3_{15},0\right)$ and hook code $$\mathfrak{Q}(P)=\left((3,1^2, 0)_{13},(5,4,1)_{14}, (2^2,1)_{15}\right).$$

In Figure \ref{determined}, we illustrate the process of decomposing $P$ into its single-block components, $P_{13}=(7^2, 5, 4^2, 3, 1^2)$, $P_{14}=(8, 6^2,4, 3, 2^2)$, and $P_{15}=(6, 5^2, 4, 2^2)$ of diagonal lengths, respectively, $T_{13}=(1, \dots, 7, 4,0)$,
$T_{14}=(1, \dots, 7, 3,0)$, and $T_{15}=(1, \dots, 6, 3,0)$.

In each part of Figure \ref{determined}, the bubbles correspond to degree-$i$ monomials. The hand monomials of $V_{i2}$ are black and the horizontal border monomials, which all belong to $V_{i1}$ are blue. For $i=15$, since $x^{15}\not \in (C_P)_{15}$, in addition of the horizontal border monomials, the set $V_{15,2}$ also includes a vertical border monomial that is illustrated in red. The rest of the bubbles which are in light grey determine the rows and columns of $P$ that need to be removed, according to Proposition \ref{rm_rows_cols_P_get_P_i}, in order to obtain the corresponding single block component.

\begin{figure}

\begin{tikzpicture}[scale=.45]

\draw[black]
	  node at (6,2.5) {Obtaining $P_{13}=(7^2,5,4^2,3,1^2)$}
	  node at (6,1.25){from $P=(15,12^4,11,7,6^2,5,3^4)$};

\draw[very thin]
	                (0,0)--(15,0)
	                (0,-1)--(15,-1)
	                (0,-2)--(12,-2)
	                (0,-3)--(12,-3)
	                (0,-4)--(12,-4)
	                (0,-5)--(12,-5)
	                (0,-6)--(11,-6)
	                (0,-7)--(7,-7)
	                (0,-8)--(6,-8)
	                (0,-9)--(6,-9)
	                (0,-10)--(5,-10)
	                (0,-11)--(3,-11)
	                (0,-12)--(3,-12)
	                (0,-13)--(3,-13)

	                (1,0)--(1,-14)
	                (2,0)--(2,-14)
	                (3,0)--(3,-10)
	                (4,0)--(4,-10)
	                (5,0)--(5,-9)
	                (6,0)--(6,-7)
	                (7,0)--(7,-6)
	                (8,0)--(8,-6)
	                (9,0)--(9,-6)
	                (10,0)--(10,-6)
	                (11,0)--(11,-6)
	                (12,0)--(12,-1)
	                (13,0)--(13,-1)
	                (14,0)--(14,-1)
	                     ;

\draw[very thick]
	(0,0)--(15,0)--(15,-1)--(12,-1)--(12,-2)--(12,-5)--(11,-5)--(11,-6)--(7,-6)--(7,-7)--(6,-7)--(6,-9)--(5,-9)--(5,-10)--(3,-10)--(3,-14)--(0,-14)--(0,0);

\draw[very thin,fill=black!5,rounded corners=2mm]
	(13,0) rectangle (14,-1)
	(10,-3) rectangle (11,-4)
	(9,-4) rectangle (10,-5)
	(8,-5) rectangle (9,-6)
	(1,-12) rectangle (2,-13)
	(0,-13) rectangle (1,-14);

\draw[very thin,fill=blue!25,rounded corners=2mm]
    (12,-1) rectangle (13,-2)
    (7,-6) rectangle (8,-7)
	(6,-7) rectangle (7,-8)
	(3,-10) rectangle (4,-11);

\draw[very thin,fill=black!50,rounded corners=2mm]
    (11,-2) rectangle (12,-3)
    (5,-8) rectangle (6,-9)
	(4,-9) rectangle (5,-10)
	(2,-11) rectangle (3,-12);
\draw[white]
    node at (0,-14.25) {space};

\draw[thick, dashed, red]
	                    (-0.5,-0.5)--(15.5,-0.5)
	                    (-0.5,-3.5)--(12.5,-3.5)
	                    (-0.5,-4.5)--(12.5,-4.5)
	                    (-0.5,-5.5)--(11.5,-5.5)
	                    (-0.5,-12.5)--(3.5,-12.5)
	                    (-0.5,-13.5)--(3.5,-13.5)

	                    (0.5,0.5)--(0.5,-14.5)
	                    (1.5,0.5)--(1.5,-14.5)
	                    (8.5,0.5)--(8.5,-6.5)
	                    (9.5,0.5)--(9.5,-6.5)
	                    (10.5,0.5)--(10.5,-6.5);

\end{tikzpicture}
\quad \quad \quad \quad

\begin{tikzpicture}[scale=.45]
\draw[black]
	    node at (6,2.5) {Obtaining $P_{14}=(8,6^2,4,3,2^2)$}
	    node at (6,1.25){from $P=(15,12^4,11,7,6^2,5,3^4)$};

	\draw[very thin]
	                (0,0)--(15,0)
	                (0,-1)--(15,-1)
	                (0,-2)--(12,-2)
	                (0,-3)--(12,-3)
	                (0,-4)--(12,-4)
	                (0,-5)--(12,-5)
	                (0,-6)--(11,-6)
	                (0,-7)--(7,-7)
	                (0,-8)--(6,-8)
	                (0,-9)--(6,-9)
	                (0,-10)--(5,-10)
	                (0,-11)--(3,-11)
	                (0,-12)--(3,-12)
	                (0,-13)--(3,-13)

	                (1,0)--(1,-14)
	                (2,0)--(2,-14)
	                (3,0)--(3,-10)
	                (4,0)--(4,-10)
	                (5,0)--(5,-9)
	                (6,0)--(6,-7)
	                (7,0)--(7,-6)
	                (8,0)--(8,-6)
	                (9,0)--(9,-6)
	                (10,0)--(10,-6)
	                (11,0)--(11,-6)
	                (12,0)--(12,-1)
	                (13,0)--(13,-1)
	                (14,0)--(14,-1)
	                     ;

    \draw[very thick]
	(0,0)--(15,0)--(15,-1)--(12,-1)--(12,-2)--(12,-5)--(11,-5)--(11,-6)--(7,-6)--(7,-7)--(6,-7)--(6,-9)--(5,-9)--(5,-10)--(3,-10)--(3,-14)--(0,-14)--(0,0);

	\draw[very thin,fill=black!5,rounded corners=2mm]
	 (12,-2) rectangle (13,-3)
	(10,-4) rectangle (11,-5)
	(9,-5) rectangle (10,-6)
	(7,-7) rectangle (8,-8)
	(6,-8) rectangle (7,-9)
	(5,-9) rectangle (6,-10)
    (3,-11) rectangle (4,-12)
	(1,-13) rectangle (2,-14);

    \draw[very thin,fill=blue!25,rounded corners=2mm]
    (13,-1) rectangle (14,-2)
    (8,-6) rectangle (9,-7)
    (5,-9) rectangle (6,-10)
	(4,-10) rectangle (5,-11)
	(0,-14) rectangle (1,-15);

    \draw[very thin,fill=black!50,rounded corners=2mm]
    (14,0) rectangle (15,-1)
    (11,-3) rectangle (12,-4)
    (2,-12) rectangle (3,-13);

\draw[thick, dashed, red]

	                    (-0.5,-2.5)--(13.5,-2.5)
	                    (-0.5,-4.5)--(12.5,-4.5)
	                    (-0.5,-5.5)--(11.5,-5.5)
	                    (-0.5,-7.5)--(8.5,-7.5)
	                    (-0.5,-8.5)--(7.5,-8.5)
	                    (-0.5,-11.5)--(4.5,-11.5)
	                    (-0.5,-13.5)--(3.5,-13.5)

	                    (1.5,0.5)--(1.5,-14.5)
	                    (3.5,0.5)--(3.5,-12.5)
	                    (6.5,0.5)--(6.5,-9.5)
	                    (7.5,0.5)--(7.5,-8.5)
	                    (9.5,0.5)--(9.5,-6.5)
	                    (10.5,0.5)--(10.5,-6.5)
	                    (12.5,0.5)--(12.5,-3.5);

 \end{tikzpicture}

 \bigskip
 \bigskip
 \bigskip

 \begin{center}
 \begin{tikzpicture}[scale=.45]
\draw[black]
        node at (6,2.5) {Obtaining $P_{15}=(6,5^2,4,2^2)$}
	    node at (6,1.25){from $P=(15,12^4,11,7,6^2,5,3^4)$};
	\draw[very thin]
	                (0,0)--(15,0)
	                (0,-1)--(15,-1)
	                (0,-2)--(12,-2)
	                (0,-3)--(12,-3)
	                (0,-4)--(12,-4)
	                (0,-5)--(12,-5)
	                (0,-6)--(11,-6)
	                (0,-7)--(7,-7)
	                (0,-8)--(6,-8)
	                (0,-9)--(6,-9)
	                (0,-10)--(5,-10)
	                (0,-11)--(3,-11)
	                (0,-12)--(3,-12)
	                (0,-13)--(3,-13)

	                (1,0)--(1,-14)
	                (2,0)--(2,-14)
	                (3,0)--(3,-10)
	                (4,0)--(4,-10)
	                (5,0)--(5,-9)
	                (6,0)--(6,-7)
	                (7,0)--(7,-6)
	                (8,0)--(8,-6)
	                (9,0)--(9,-6)
	                (10,0)--(10,-6)
	                (11,0)--(11,-6)
	                (12,0)--(12,-1)
	                (13,0)--(13,-1)
	                (14,0)--(14,-1)
	                     ;

    \draw[very thick]
	(0,0)--(15,0)--(15,-1)--(12,-1)--(12,-2)--(12,-5)--(11,-5)--(11,-6)--(7,-6)--(7,-7)--(6,-7)--(6,-9)--(5,-9)--(5,-10)--(3,-10)--(3,-14)--(0,-14)--(0,0);

	\draw[very thin,fill=black!5,rounded corners=2mm]
	 (13,-2) rectangle (14,-3)
	(6,-9) rectangle (7,-10)
    (4,-11) rectangle (5,-12)
	(12,-3) rectangle (13,-4)
	(8,-7) rectangle (9,-8)
	(7,-8) rectangle (8,-9)
	(6,-9) rectangle (7,-10)
	(5,-10) rectangle (6,-11)
	(3,-12) rectangle (4,-13)
	(0,-15) rectangle (1,-16);

    \draw[very thin,fill=blue!25,rounded corners=2mm]
    (14,-1) rectangle (15,-2)
    (9,-6) rectangle (10,-7)
	(1,-14) rectangle (2,-15);

    \draw[very thin,fill=black!50,rounded corners=2mm]
    (2,-13) rectangle (3,-14)
    (11,-4) rectangle (12,-5)
	(10,-5) rectangle (11,-6);

	\draw[very thin,fill=red!25,rounded corners=2mm]
	(15,0) rectangle (16,-1);

\draw[thick, dashed, red]

	                    (-0.5,-2.5)--(14.5,-2.5)
	                    (-0.5,-3.5)--(13.5,-3.5)
	                    (-0.5,-7.5)--(9.5,-7.5)
	                    (-0.5,-8.5)--(8.5,-8.5)
	                    (-0.5,-9.5)--(7.5,-9.5)
	                    (-0.5,-10.5)--(6.5,-10.5)
	                    (-0.5,-11.5)--(5.5,-11.5)
	                    (-0.5,-12.5)--(4.5,-12.5)

	                    (0.5,0.5)--(0.5,-16.5)
	                    (3.5,0.5)--(3.5,-13.5)
	                    (4.5,0.5)--(4.5,-12.5)
	                    (5.5,0.5)--(5.5,-11.5)
	                    (6.5,0.5)--(6.5,-10.5)
	                    (7.5,0.5)--(7.5,-9.5)
	                    (8.5,0.5)--(8.5,-8.5)
	                    (12.5,0.5)--(12.5,-4.5)
	                    (13.5,0.5)--(13.5,-3.5);

 \end{tikzpicture}
 \end{center}
         \caption{Finding single-block components for partition $P=(15,12^4,11,7,6^2,5,3^4)$ of Example \ref{1stgensEx} using Proposition \ref{rm_rows_cols_P_get_P_i}. In each part of the Figure, the black bubbles represent elements of $V_{i2}$, the blue bubbles represent horizontal border monomials in $V_{i1}$ and the red bubble represents the vertical border monomial in $V_{i1}$ (when $V_{i1}$ includes one such monomial).}\label{determined}

\end{figure}

\end{example}

\subsubsection{The projection map $\pi$ from ${\mathbb V}(E_P)$ to a product of cells of  ``small'' Grassmannians.}
\begin{definition}[The map $\pi$ of $\mathbb V(E_P)$ to the product of small Grassmannians]\label{def_pi}\footnote{We use the term ``small'' Grassmannian to distinguish these from the (large) Grassmannians determined by the projection $\G_T\to \prod_{d\le i\le {\sf j}} \Grass(i+1-t_i, i+1)$ given by $I\to (I_d,\ldots, I_{\sf j})$.}

Suppose $P\in \mathcal P(T)$ and let $I\in \mathbb V(E_P)$. Denote by $W_i$ the vector space generated by $V_{i1}\cup (C_P)_i$ for $i\in [d,{\sf j}]$. Note that $\dim_kW_i=\delta_i+1+t_i$.
It is straightforward to see that the vector space $I_{W_i}=I\cap W_i$ has dimension $(\delta_i+1)$. The leading monomial of any non-zero element of $I_{W_i}$ belongs to $V_{i1}$ and conversely, given an element $M$ of $V_{i1}$, there is an element of $I_{W_i}$ whose leading monomial is $M$.\\
Let $K_i$ be the vector space generated by $V_{i3}=(C_P)_i\setminus V_{i2}$. We write $W_i$ as a direct sum $W_i=V_i\oplus K_i$ and consider the projection on the first factor
$\displaystyle{pr_1: W_i\longrightarrow V_i }$ and let $I_{V_i}=pr_1(I_{W_i})$. Then $I_{V_i}\in \Grass(\delta_i+1, V_i)$.
\\
Thus, we have constructed a morphism
${\mathbb V}(E_P) \stackrel{\pi_i}{\longrightarrow} \Grass(\delta_i+1, V_i)$,
and, taking the product $\pi=(\pi_d,\ldots, \pi_{\sf j})$
we have a morphism
\begin{equation}\label{isomeq}
{\mathbb V}(E_P) \stackrel{\pi}{\longrightarrow} \prod_{i=d}^{i={\sf j}}\Grass(\delta_i+1, V_i).
\end{equation}
\end{definition}
\begin{remark}
Note that by construction and the definition of the difference-one hook code, the image of $\pi_i$ is a Schubert cell $\mathbb V(E_{P_i})$ in
$\Grass(\delta_i+1, V_i)$,
whose dimension is $\left|{\mathcal Q}(P)_i\right|$, the length of the $i$-th block of the hook code of $P$. \\
When $\delta_{i+1}=0$ (that is, $t_i=t_{i+1}$) we have $\dim(V_{i})=\delta_i+1$ and $\Grass(\delta_i+1, V_i)$ is just one point: the
$i$-th block of the hook code of $P$ in this case is empty, so has length zero (see Remark \ref{empty_hook_code_block_partition}).
\end{remark}
\begin{lemma}[Morphism $\pi$ to the component small Grassmannians]\label{morphismlem} Let $P\in \mathcal P(T)$. The morphism $\pi$ of \eqref{isomeq} determines a morphism $$\pi:{\mathbb V}(E_P) \stackrel{\pi}{\longrightarrow} \prod_{i=d}^{i={\sf j}}\mathbb V(E_{P_i}).$$
\end{lemma}
\begin{proof} The bijection $s_i : V_i\rightarrow\Mon(R_{d_i})$ of Equation \eqref{monomeq} induces a linear isomorphism
 $\tilde{s_i}:V_i\to R_{d_i}$. We thus have an isomorphism $\tilde{s_i}:\Grass(\delta_i+1, V_i)\to \Grass(\delta_i+1, R_{d_i})$, taking
 $I_{V_i}=pr_1(I_{W_i})$ to the subspace $\tilde{s_i} (I_{V_i})\subset R_{d_i}$. Then by Definition \ref{Pidef1}, Equation \ref{isomeq} determines the morphism $\pi$ of the Lemma.
\end{proof}

\begin{theorem}\label{projectionthm}The morphism $\pi$ of
Lemma \ref{morphismlem} is an isomorphism from $\mathbb V(E_P)$ onto its image $\prod_{i=d}^{i={\sf j}}\mathbb V(E_{P_i})$.
\end{theorem}
\begin{proof} Each $\mathbb V(E_{P_i})$ is an affine space; from Theorem \ref{GTdecompthm} and Lemma \ref{hookP(i)lem} the dimension of $\mathbb V(E_P)$ is the sum $\sum \dim \mathbb V(E_{P_i})$.
We know that the difference-one hook code of $P_i$ is the $i$-$th$ component $\big(\mathfrak{h}_i(P)\big)$.
\\
Using the notation of Definition \ref{Pidef1}, consider the set $V_{i1}$, which contains all the degree-$i$ horizontal border monomials of $E_P$.
\\Let $V_{i1}=\left\{M_{i,1}, \dots , M_{i,\delta_i}, M_{i,{\delta_i+1}} \right\}$ (numbered from top to bottom). Denote by $b_{i,l}$ the number of degree-$i$ hand monomials above $M_{i,l}$. Then we have $b_{i,{\delta_i+1}}\geq b_{i,\delta_i}\geq \cdots \geq b_{i,1}$, and the affine space $\mathbb V(E_{P_i})$ has dimension $\displaystyle{\sum_{l=1}^{l=\delta_i+1}b_{i,l}=\left|\big(\mathfrak{h}_i(P)\big)\right|}$. We may think of $\mathbb V(E_{P_i})$ as
$\displaystyle{\prod_{l=1}^{l=\delta_i+1}{\sf k}^{b_{i,l}}}$ (\textsl{$b_{i,l}$ free parameters} for each $M_{i,l}$).
We now let $\hMon (E_P)$ be the set of horizontal border monomials of $E_P$.
Suppose $\hMon (E_P)=\left\{y^{\beta_0}, xy^{\beta_1}, \cdots , x^my^{\beta_m} \right \}$, with $\beta_0\geq \beta_1 \geq \cdots \geq \beta_m$. \\
For $0\leq l \leq m$, let $b_l$ be the number of degree-$(l+\beta_l)$ hand monomials above $x^ly^{\beta_l}$. \\
Let $P^{\prime}$ be the partition obtained by deleting the first column of $P$.
The morphism ${\mathbb V}(E_P) \rightarrow {\mathbb V}(E_{P^{\prime}}): I\mapsto (I:x)$ is a trivial fibration whose fiber has dimension $b_0$ and we have
${\mathbb V}(E_P) \cong {\sf k}^{b_0}\times {\mathbb V}(E_{P^{\prime}})$ (see for example \cite{Y0}, Prop. 1.7).  We then obtain the proof of the Theorem by induction.
\end{proof}

\subsubsection{Examples of the projection map $\pi$.}
\begin{example} Let $T=(1,2,2,1)$.  We have $\delta_2=t_1-t_2=0$, $\delta_3=t_2-t_3=1$,
$\delta_4=t_3-t_4=1$, and $\mathfrak B(T)=\left(\mathfrak B_2,\mathfrak B_3\right)=\left((1\times 1)_2,(1\times 2)_3\right)$.
Here the product of  ``small'' Grassmannians is
$$G=\Grass(\delta_2+1, \delta_2+1+\delta_3)\times \Grass(\delta_3+1, \delta_3+1+\delta_4)=\Grass( 1,2)\times \Grass(2,3).$$
Consider the partition $P=(3,3)$.
\begin{center}
$P: \ydiagram{3,3}.$
\end{center}
By Definition \ref{def_pi}, we have $V_2=\langle y^2,x^2\rangle$,  $W_2=\langle y^2,xy, x^2\rangle$, $ V_3=\langle xy^2,x^2y,x^3\rangle$ and $ W_3=V_3$. \\
Any element $I$ in the cell ${\mathbb V}(E_P)$ is of the form $I=(y^2+a_1xy+a_2x^2,x^3)$, with $(a_1,a_2)\in {\sf k}^2$. So ${\mathbb V}(E_P)$
is a two dimensional affine space. \\
For $I=(y^2+a_1xy+a_2x^2,x^3)\in {\mathbb V}(E_P)$ we get $I\cap W_2=\langle y^2+a_1xy+a_2x^2\rangle$ and the projection of $I\cap W_2$ on $V_2$ is $I_{V_2}=\langle y^2+a_2x^2\rangle \in \Grass(1,V_2)$.
 Also $I\cap W_3=\langle xy^2+a_1x^2y,x^3\rangle=I_{V_3}\in \Grass(2, V_3)$.
We view ${\mathbb V}(E_P)$ as a product of two cells, one in $\Grass( 1,2)$ corresponding to  single block $T_2=(1,1)$ and another one in $\Grass(2,3)$ corresponding to $T_3=(1,2,1)$. \\
The difference-one hook code of $P=(3,3)$ is $${\mathfrak Q}(P)=((1)_2,(1)_3)\subset \mathcal B(T)=((1\times 1)_2, (1\times 2)_3).$$ The code $(1)_2$ corresponds to the vector space $V_2$ and the small cell $\mathbb V(E_{P_2})=\left\{ \langle y^2+a_2x^2\rangle,{a_2\in {\sf k}}\right\}\subset \Grass( 1,2)\cong\Grass(1,V_2)$:
\begin{center}
\begin{tabular}{c}
$P_2$ \\ \ydiagram{2}
\end{tabular}
.
\end{center}
 The code $(1)_3$ corresponds to the vector space $V_3$ and the small cell $\mathbb V(E_{P_3})=\left\{ \langle xy^2+a_1x^2y,x^3\rangle,{a_1\in {\sf k}}\right\}\subset\Grass(2,3)\cong\Grass(2,V_3)$ (note, these are labelled by degree: $V_i\subset R_i$):
 \begin{center}
\begin{tabular}{c}
$P_3$ \\ \ydiagram{2,2}
\end{tabular}
.
\end{center}

So we get ${\mathbb V}(E_P)$ as a product of two affine lines: ${\mathbb V}(E_P)=\mathbb V(E_{P_2})\times \mathbb V(E_{P_3})$. \end{example}
\noindent
\begin{example}\label{53313ex} Let $T=(1,2,3,4,5,4,2)$ and consider $P=(5^3,3,1^3)$.  (See Figure \ref{fig5}.)
We have $(\delta_5,\delta_6,\delta_7)=(1,2,2)$. The two single-block Hilbert functions associated to $T$ are $T_5=(1,2,3,2)$ and $T_6=(1,2,3,4,2)$
(see Definition \ref{Pidef1}). \\
We will view the cell ${\mathbb V}(E_P)$ as a product of cells in
$\Grass( 2,4)\times \Grass(3,5)$. \\
The difference-one hook code of $P$ is ${\mathfrak Q}(P)=\big((1,1)_5,(2,0)_6\big)$. \\
The partition $P_5=(3^2,2)$ of diagonal lengths $T_5$ has hook code ${\mathfrak Q}(P_5)=\big( (1,1)\big)$ (the degree-$5$ block of ${\mathfrak Q}(P)$).
The partition $P_6=(4^2,2,1^2)$ of diagonal lengths $T_6$ has hook code ${\mathfrak Q}(P_6)=\big( (2,0)\big)$ (the degree-$6$ block of ${\mathfrak Q}(P)$).
\begin{figure}[!h]
\begin{center}
	\begin{tikzpicture}[scale=.8]
	\draw[dashed,very thin] 	(0,0)--(5,0)
						(0,-1)--(5,-1)
						(0,-2)--(5,-2)
						(0,-3)--(3,-3)
						(0,-4)--(1,-4)
						(0,-5)--(1,-5)
						(0,-6)--(1,-6)
						(0,0)--(0,-7)
						(1,0)--(1,-4)
						(2,0)--(2,-4)
						(3,0)--(3,-3)
						(4,0)--(4,-3);
	\draw[thin, red,dashed,fill=yellow!10] (4,-1) rectangle (5,-2)
							 (2,-3) rectangle (3,-4)
							 (4,-2) rectangle (5,-3)
							 (0,-6) rectangle (1,-7);
	\draw[thin, blue,fill=black!5,rounded corners=3mm] (1,-4) rectangle (2,-5)
											    (2,-4) rectangle (3,-5)
											    (3,-3) rectangle (4,-4)
											    (5,0) rectangle (6,-1);
	\draw[red] node at (4.5,-1.5) {\footnotesize{$x^4y$}}
			  node at (2.5,-3.5) {\footnotesize{$x^2y^3$}}
			  node at (4.5,-2.5) {\footnotesize{$x^4y^2$}}
			  node at (0.5,-6.5) {\footnotesize{$y^6$}};
	\draw[blue] node at (1.5,-4.5) {\footnotesize{$xy^4$}}
			    node at (2.5,-4.5) {\footnotesize{$x^2y^4$}}
			    node at (3.5,-3.5) {\footnotesize{$x^3y^3$}}
			    node at (5.5,-0.5) {\footnotesize{$x^5$}};
	\draw[very thick] (5,0)--(5,-3)--(3,-3)--(3,-4)--(1,-4)--(1,-7)--(0,-7);
         \end{tikzpicture}
         \caption{Ferrers diagram of $P=(5^3,3,1^3)$: \textsl{hand monomials} are marked in red and
         \textsl{border monomials} are {marked} in blue (Example \ref{53313ex}).}\label{fig5}
\end{center}
\end{figure}
\\
We now display the isomorphism $\pi: {\mathbb V}(E_P)\to{\mathbb V}(E_{P_5})\times {\mathbb V}(E_{P_6})$. \\

By Definition \ref{def_pi}, we have
$$V_5=\langle xy^4,x^2y^3,x^4y,x^5 \rangle, W_5=R_5, V_6=\langle y^6,x^2y^4,x^3y^3,x^4y^2,x^5y\rangle=W_6.$$
The projection of $I_5$ onto $V_5$ is a $2$-dimensonial vector space $$I_{V_5}=\langle xy^4+\alpha_1x^2y^3+\alpha_2x^4y,x^5 \rangle ,
(\alpha_1,\alpha_2)\in {\sf k}^2.$$
The projection of $I_6$ onto $V_6$ is a $3$-dimensonial vector space $$I_{V_6}=\langle x^2y^4+a_1x^4y^2,x^3y^3+a_2x^4y^2,x^5y \rangle ,
(a_1,a_2)\in {\sf k}^2.$$
So we have $I_{V_5}\in \Grass(2,V_5)\cong \Grass(2,4)$, $I_{V_5}\in \Grass(3,V_6)\cong \Grass(3,5)$ and ${\mathbb V}(E_P)$ can be viewed as ${\mathbb V}(E_{P_5})\times {\mathbb V}(E_{P_6})$. \vskip 0.2cm\noindent
\textbf{Note.} Suppose we are given $(L_5,L_6)\in {\mathbb V}(E_{P_5})\times {\mathbb V}(E_{P_6})$ with
$L_5=\langle xy^4+\alpha_1x^2y^3+\alpha_2x^4y,x^5 \rangle , (\alpha_1,\alpha_2)\in {\sf k}^2$ and
$L_6=\langle x^2y^4+a_1x^4y^2,x^3y^3+a_2x^4y^2,x^5y \rangle , (a_1,a_2)\in {\sf k}^2$. Then, using standard basis techniques, one can see that there
is a unique ideal $I\in {\mathbb V}(E_P)$ such that $I_{V_5}=L_5$ and $I_{V_6}=L_6$:
$$
I=\left(x^5, x^3y^3+a_2x^4y^2, x((y^4+a_1x^2y^2)+\alpha_1(xy^3+a_2x^2y^2)+\alpha_2x^3y), y^7\right).
$$
\end{example}

\begin{example}\label{972ex}
Let $T=(1,2,3,4,5,6,7,5,2,0)$ and consider the partition $P=(9,7^2,4^2,2,1^2)$. (See Figure \ref{fig6}). We have $( \delta_7,\delta_8,\delta_9)=(2,3,2)$, $T_7=(1,2,3,4,5,3)$ and $T_8=(1,2,3,4,5,2)$.\\
The product of small Grassmannians here is
$$G=\Grass(\delta_7+1, \delta_7+1+\delta_8)\times \Grass(\delta_8+1, \delta_8+1+\delta_9)=\Grass(3,6)\times \Grass(4,6).$$
\begin{figure}[!h]
\begin{center}
	\begin{tikzpicture}[scale=.8]
	\draw[dashed,very thin] 	(0,0)--(9,0)
						(0,-1)--(7,-1)
						(0,-2)--(7,-2)
						(0,-3)--(4,-3)
						(0,-4)--(4,-4)
						(0,-5)--(2,-5)
						(0,-6)--(1,-6)
						(0,-7)--(1,-7)
						(0,0)--(0,-8)
						(1,0)--(1,-6)
						(2,0)--(2,-5)
						(3,0)--(3,-5)
						(4,0)--(4,-3)
						(5,0)--(5,-3)
						(6,0)--(6,-3)
						(7,0)--(7,-1)
						(8,0)--(8,-1);
	\draw[thin, dashed,red,fill=yellow!10] (8,0) rectangle (9,-1)
							(6,-1) rectangle (7,-2)
							(6,-2) rectangle (7,-3)
							(3,-4) rectangle (4,-5)
							(0,-7) rectangle (1,-8);
	\draw[thin, blue,fill=black!5,rounded corners=3mm] (7,-1) rectangle (8,-2)
						     (5,-3) rectangle (6,-4)
						     (4,-3) rectangle (5,-4)
						     (3,-5) rectangle (4,-6)
						     (2,-5) rectangle (3,-6)
						     (1,-6) rectangle (2,-7)
						     (0,-8) rectangle (1,-9);
	\draw[red] node at (8.5,-0.5) {\footnotesize{$x^8$}}
			  node at (6.5,-1.5) {\footnotesize{$x^6y$}}
			  node at (6.5,-2.5) {\footnotesize{$x^6y^2$}}
			  node at (3.5,-4.5) {\footnotesize{$x^3y^4$}}
			  node at (0.5,-7.5) {\footnotesize{$y^7$}};
	\draw[blue] node at (7.5,-1.5) {\footnotesize{$x^7y$}}
			    node at (5.5,-3.5) {\footnotesize{$x^5y^3$}}
			    node at (4.5,-3.5) {\footnotesize{$x^4y^3$}}
			    node at (3.5,-5.5) {\footnotesize{$x^3y^5$}}
			    node at (2.5,-5.5) {\footnotesize{$x^2y^5$}}
			    node at (1.5,-6.5) {\footnotesize{$xy^6$}}
			    node at (0.5,-8.5) {\footnotesize{$y^8$}};
	\draw[very thick] (9,0)--(9,-1)--(7,-1)--(7,-3)--(4,-3)--(4,-5)--(2,-5)--(2,-6)--(1,-6)--(1,-8)--(0,-8);
         \end{tikzpicture}
         \caption{Ferrers diagram of $P=(9,7^2,4^2,2,1^2)$: \textsl{border monomials} are marked in blue and
         \textsl{hand monomials} are marked in red (Example \ref{972ex}).}\label{fig6}
\end{center}
\end{figure}\\
The difference-one hook code of $P$ is ${\mathfrak Q}(P)=\big((3,2,0)_7,(4,3)_8\big)$. \\
\begin{enumerate}[(a)]
\item In degree $7$ we have \textcolor{blue}{$V_{7,1}=\left\{x^4y^3,x^2y^5,xy^6\right\}$},
\textcolor{red}{$V_{7,2}=\left\{x^6y,x^3y^4,y^7\right\}$}, so
$V_7$ has basis $(x^6y,x^4y^3,x^3y^4,x^2y^5,xy^6,y^7)$ in the lex order.
We consider the bijection $s_7: V_{7,1}\cup V_{7,2} \longrightarrow \Mon(R_{5})$ given by:
$$
\begin{array}{ccc}
\textcolor{red}{s_7(x^6y)=x^5}, &\textcolor{blue}{s_7(x^4y^3)=x^4y} ,  &  \textcolor{red}{s_7(x^3y^4)=x^3y^2} \\
\textcolor{blue}{s_7(x^2y^5)=x^2y^3}, & \textcolor{blue}{s_7(xy^6)=xy^4}, & \textcolor{red}{s_7(y^7)=y^5}.
\end{array}
$$
We then get a one block partition $P_7$ (see Figure \ref{fig7})
\begin{figure}[!h]
\begin{center}
	\begin{tikzpicture}[scale=.8]
	\draw[dashed,very thin] 	(0,0)--(6,0)
						(0,-1)--(4,-1)
						(0,-2)--(4,-2)
						(0,-3)--(2,-3)
						(0,-4)--(1,-4)
						(0,-5)--(1,-5)
						(0,0)--(0,-6)
						(1,0)--(1,-4)
						(2,0)--(2,-3)
						(3,0)--(3,-3)
						(4,0)--(4,-1)
						(5,0)--(5,-1);
	\draw[thin, dashed,red,fill=yellow!10] (5,0) rectangle (6,-1)
								     (3,-2) rectangle (4,-3)
								     (0,-5) rectangle (1,-6);
	\draw[thin, blue,fill=black!5,rounded corners=3mm] (4,-1) rectangle (5,-2)
											   (2,-3) rectangle (3,-4)
											   (1,-4) rectangle (2,-5);
	\draw[red] node at (5.5,-0.5) {\footnotesize{$x^5$}}
			  node at (3.5,-2.5) {\footnotesize{$x^3y^2$}}
			  node at (0.5,-5.5) {\footnotesize{$xy^4$}};
	\draw[blue] node at (4.5,-1.5) {\footnotesize{$x^4y$}}
			    node at (2.5,-3.5) {\footnotesize{$x^2y^3$}}
			    node at (1.5,-4.5) {\footnotesize{$xy^4$}};
	\draw[very thick] (6,0)--(6,-1)--(4,-1)--(4,-3)--(2,-3)--(2,-4)--(1,-4)--(1,-6)--(0,-6);
         \end{tikzpicture}
         \caption{Ferrers diagram of $P_7=(6,4^2,2,1^2)$ : \textsl{horizontal-border monomials} are marked in blue and
         \textsl{hand monomials} are marked in red (Example \ref{972ex}a).}\label{fig7}
\end{center}
\end{figure}

\item In degree $8$, \textcolor{blue}{$V_{8,1}=\left\{x^7y,x^5y^3,x^3y^5,y^8\right\}$},
\textcolor{red}{$V_{8,2}=\left\{x^8,x^6y^2\right\}$}.
$V_8$ has basis $(x^8,x^7y,x^6y^2,x^5y^3,x^3y^5,y^8)$ in the lex order. \\
The bijection $s_8: V_{8,1}\cup V_{8,2} \longrightarrow\Mon(R_{5})$ given by:
$$
\begin{array}{ccc}
\textcolor{red}{s_8(x^8)=x^5}, & \textcolor{blue}{s_8(x^7y)=x^4y},  &  \textcolor{red}{s_8(x^6y^2)=x^3y^2}\\
\textcolor{blue}{s_8(x^5y^3)=x^2y^3}, & \textcolor{blue}{s_8(x^3y^5)=xy^4}, & \textcolor{blue}{s_8(y^8)=y^5}.
\end{array}
$$
This gives us a one block partition $P_8$ (see Figure \ref{fig8})
\begin{figure}[!h]
\begin{center}
	\begin{tikzpicture}[scale=.8]
	\draw[dashed,very thin] 	(0,0)--(6,0)
						(0,-1)--(4,-1)
						(0,-2)--(4,-2)
						(0,-3)--(2,-3)
						(0,-4)--(1,-4)
						(0,0)--(0,-5)
						(1,0)--(1,-4)
						(2,0)--(2,-3)
						(3,0)--(3,-3)
						(4,0)--(4,-1)
						(5,0)--(5,-1);
	\draw[thin, dashed,red,fill=yellow!10] (5,0) rectangle (6,-1)
								     (3,-2) rectangle (4,-3);
	\draw[thin, blue,fill=black!5,rounded corners=3mm] (4,-1) rectangle (5,-2)
											   (2,-3) rectangle (3,-4)
											   (1,-4) rectangle (2,-5)
											   (0,-5) rectangle (1,-6);
	\draw[red] node at (5.5,-0.5) {\footnotesize{$x^5$}}
			  node at (3.5,-2.5) {\footnotesize{$x^3y^2$}};
	\draw[blue] node at (4.5,-1.5) {\footnotesize{$x^4y$}}
			    node at (2.5,-3.5) {\footnotesize{$x^2y^3$}}
			    node at (1.5,-4.5) {\footnotesize{$xy^4$}}
			    node at (0.5,-5.5) {\footnotesize{$y^5$}};
	\draw[very thick] (6,0)--(6,-1)--(4,-1)--(4,-3)--(2,-3)--(2,-4)--(1,-4)--(1,-5)--(0,-5);
         \end{tikzpicture}
         \caption{Ferrers diagram $P_8=(6,4^2,2,1)$ : \textsl{horizontal-border monomials} are marked in blue and
         \textsl{hand monomials} are marked in red (Example \ref{972ex}b).}\label{fig8}
\end{center}
\end{figure}
\end{enumerate}

\end{example}

\section{Number of generators for a single-block partition.}\label{singlekappasection}

We first state the known bounds for the number of generators of a graded ideal $I$ of Hilbert function $H(R/I)=T$ for arbitrary $T$ (Lemma \ref{kappaTlem}).
In Theorem \ref{kappathm} we determine the number of generators $\kappa(P)$ for generic ideals in the cell $\mathbb V(E_P)$ where $P$ has diagonal lengths $T$ satisfying the single-block Equation \eqref{Tsingleeqn}.
\subsection{Lower bound $\kappa(T)$ on the number of generators of an ideal $I$ in $\G_T$.}
We recall Equation \ref{Teq} for an arbitrary codimension two Hilbert function $T$:
\begin{equation*}
T=(1,2,\ldots, d-1,d,t_{d},\ldots, t_{\sf j},0) \text { where } d\ge t_d\ge t_{d+1}\ge \cdots \ge t_{\sf j}>0.
\end{equation*}
Here ${\sf j}$ is the (highest) socle degree of $A=R/I$. Recall from Section \ref{GTsec} that $\G_T$ is the irreducible projective variety parametrizing the graded ideals $I$ in $R={\sf k}[x,y]$ such that $A=R/I$ has Hilbert function $T$.
\begin{definition}[Order of a Hilbert function $T$]\label{order_of_Tdef}
Let $T$ be a sequence satisfying Equation \eqref{Teq}. Set $\nu(T)=d$, usually called the order of $T$: that is $\nu(T)$ is the order of graded ideals $I\in \G_T$ -- that define an Artinian algebra $A=R/I$ of Hilbert function $T$.
\end{definition}

\begin{definition}\label{kappadef} We let $\kappa(T)$ be the minimum number of generators for the ideal $I$ corresponding to a generic element of $\G_T$.
If $P$ is a partition of diagonal lengths $T$, we denote by $E_P$ the monomial ideal associated to $P$ and set $\kappa(P)$ to be the minimum number of generators for a generic element $I$ in the cell $\mathbb{V}(E_P)$.
\end{definition}

Given a sequence $T$ satisfying Equation \eqref{Teq}, recall that we denote by $\delta_i$ the first difference function of $T$:
\begin{equation}\label{diffeq}
\delta_i=t_{i-1}-t_{i}  \text { for }  i \in [\nu(T),{\sf j}+1].
\end{equation}
\par
The following result (i)-(ii) is shown in \cite[Theorem 4.3, Lemma 4.5]{Ia}, but a separate proof will also result from
our work here (see Remark \ref{lower_bound_for_degree_i+1_gens}).  A different proof of (ii) is given by M. Mandal and M.E. Rossi in \cite[Theorem~2.1]{MR}. The statement (iii) is obvious.
We denote by $[k]^+=\max\{k,0\}$.
\begin{lemma}\label{kappaTlem} Let $T$ satisfy Equation \eqref{Teq}, and let $I$ be a homogeneous ideal such that $A=R/I$ has Hilbert function $T$. Then
\begin{enumerate}[(i).]
\item  $I$ has at least  $[\delta_i-\delta_{i-1}]^+$ generators of each degree $i\ge \nu(T)$.
\item  A generic graded ideal $I\in\G_T$ has
\begin{equation}\label{kappaTeq}
\kappa(T)=1+\delta_{\nu(T)}+\sum_{i> \nu(T)}[\delta_{i+1}-\delta_i]^+
\end{equation}
generators, exactly $[\delta_i-\delta_{i-1}]^+$ in each degree $i\ge \nu(T)$.

\item  The ideal $E_{P_0}, P_0=T$ (listed as a partition) has $\nu(T)+1=1+\sum_{i\ge \nu(T)} \delta_i$ generators, and $\kappa_{E_{P_0},i}=\delta_i$ for $i> \nu(T)$ and $1+\delta_{\nu(T)}$ for $i=\nu(T)$. This is the termwise maximum $\kappa_I(z)$ that occurs for any ideal $I\in \G_T$: that is $\kappa_{I,i}\le \delta_i$ for $i>\nu(T)$ and $\kappa_{I,\nu(T)}\le 1+\delta_{\nu(T)}$.
\end{enumerate}
\end{lemma}
\begin{definition}\label{special-partition-definition}
If $P$ is a partition of diagonal lengths $T$ such that $\kappa(P)\neq \kappa(T)$, then we say $P$ is \emph{special}.
If $\kappa(P) = \kappa(T)$ we say $P$ is \emph{non-special}.
\end{definition}\par\noindent
\begin{example}\label{12321ex} Let $T=(1,2,3,2,1)$. We have $\kappa (T)=2$, as the generic ideal in $\G_T$ is a complete intersection of generator degrees $(3,3)$. For $P=(5,3,1)$ we also have $\kappa (P)=2$: here $\mathbb V(E_P)$ is open dense in $\G_T$, so $P=(5,3,1)$ is non-special. But for $P=(3,3,1^3)$ we have $\kappa(P)=3$ since an $R$-relation between the generators $y^5,y^2x+\cdots$ cannot yield the generator $x^3$: so $P=(3,3,1^3)$ is special.
\end{example}
\subsection{Single block partitions $P$, and $\kappa(P)$.}\label{single block-partition-and-kappa-section}
Henceforth in this section we let $T$ be a Hilbert function that satisfies
\begin{equation}\label{Tsingleeqn}
T=(1,2,\ldots, d-1, d,t_d,0).
\end{equation}
where $d\ge t_d$ and we let $s=d+1-t_d$. We term this a \emph{single-block} Hilbert function.
 In this case, $\G_T$ is isomorphic to the Grassmannian variety $\Grass(s,R_d)$ where $R_d$ is the vector space of the degree $d$ homogeneous forms of $R={\sf k}[x,y]$:
$$
\begin{array}{rcl}
\Phi: \G_T & \rightarrow & \Grass(s, R_d)\\
I & \mapsto & I_d
\end{array}
.
$$
Also, by Equation \ref{kappaTeq} we have for a single-block Hilbert function
\begin{equation}\label{kappaTsingleeq}
\kappa (T)=s+\delta, \text { where }\delta=\max\{t_d+1-s,0\}.
\end{equation}
Let $P$ be a partition of diagonal lengths $T$. The corners of the Ferrers diagram of $P$ correspond to monomials $x^{\alpha}y^{\beta}$ that belong to a minimal set of generators for the monomial ideal $E_P$. We may call such monomials,  corner-monomials of $P$.
Let $I\in \G_T$ be an element of the cell $\mathbb{V}(E_P)$. Then the corner-monomials of $P$ are leading terms of a system of generators
$\mathfrak{B}(I)$ of $I$. The system $\mathfrak{B}(I)$ may not be minimal.
By definition of $T$, a minimal set of generators of $E_P$ must contain $s$ degree $d=\nu(T)$ corner-monomials. These degree $d$ corner-monomials are
leading monomials for the degree $d$ elements of the system of generators $\mathfrak{B}(I)$. Since we are looking for a minimal set of generators for $I$, we want a criterion to decide that a degree $d+1$ element of $\mathfrak{B}(I)$ can be obtained using a relation involving degree $d$ elements of
$\mathfrak{B}(I)$. That is where corner ``kick-off'' comes into play.

\medskip
Let $a$ be integer such that $0\leq a <d=\nu(T)$ and set $d^{\prime}=d-a$. Suppose $m$ is an integer such that $1<m\leq  d^{\prime}$.
For any integer $i$ such that $0\leq i\leq m$, set $\displaystyle{K_i=x^{a}\cdot\left(x^{m-i}y^{d^{\prime}-m+i}\right)}$.
The $K_i$'s form a set of $m+1$ consecutive degree $d$ monomials in two variables:
$$
x^{a+m}y^{d^{\prime}-m},x^{a+m-1}y^{d^{\prime}-m+1}, \ldots , x^{a+1}y^{d^{\prime}-1},x^{a}y^{d^{\prime}}
.
$$
From these $m+1$ consecutive monomials we have $m$ relations: $yK_i-xK_{i+1}=0$, $(0\leq i <m)$. Suppose the $K_i$'s are leading monomials of
some elements of $\mathfrak{B}(I)$. In order to produce $m$ degree $d+1$ elements of $\mathfrak{B}(I)$ with these relations, we have to assume that $2(m+1)\leq d^{\prime}+2=\dim_{{\sf k}}(R_{d^{\prime}+1})$, that is, $2m\leq d^{\prime}$. \\
Note that by definition, if $f_0, \ldots, f_m$ are degree-$i$ forms such that $f_i$ has leading monomial $K_i$, then $x^a$ divides any element of the ideal
$I=(f_0, \ldots, f_m)$. Therefore, for simplicity we now assume $a=0$, so $\displaystyle{K_i=x^{m-i}y^{d-m+i}}$,  $2m\leq d$, and we let $N_1, \ldots , N_m$ be the $m$ degree $d+1$ monomials given by
$$
\left\{
\begin{array}{l}
N_i=x^{\alpha_i}y^{\beta_i}, \ \alpha_i+\beta_i=d+1,
\\
0\leq \beta_1<\beta_2< \cdots <\beta_m<d-m
\\
m+1<\alpha_m<\alpha_{m-1}< \cdots <\alpha_1\leq d+1
\end{array}
\right.
.
$$\par
Concerning the next Lemma, although J.~Brian\c{c}on and A.~Galligo state their standard basis result that we use in characteristic zero, it is valid also for characteristic bigger than the socle degree $d$.
 This is the key step in the paper where we need to restrict the characteristic of ${\sf k}$.

\begin{lemma}[How to kick off corners]\label{corner-kick-off-lemma}
With the above notation, there exist $m+1$ degree $d$ forms $f_0, \ldots, f_m$ such that $f_i$ has leading monomial $K_i$
and $N_i$ is a leading monomial of a degree $d+1$ element of the ideal $I=(f_0, \ldots, f_m)$.
\end{lemma}
\begin{proof}
Using a technique of standard basis calculations developed by J.~Brian\c{c}on and A.~Galligo in \cite{brian-gal}, we can inductively  construct $f_0, \ldots, f_m$ such that $N_i\in I=(f_0, \ldots, f_m)$.  Let $$f_0=x^my^{d-m}, \ f_1=x^{m-1}y^{d-m+1}+\lambda_1x^{\alpha_1-1}y^{\beta_1}.$$
One can see that $xf_1-yf_0=\lambda_1x^{\alpha_1}y^{\beta_1} $, so if we set $\lambda_1\neq 0$, we have
$$N_1=x^{\alpha_1}y^{\beta_1}\in (f_0, \ldots, f_m).$$
In general, for $0\leq i<m$, suppose that we have $f_i=x^{m-i}u_i+\lambda_ix^{\alpha_i-1}y^{\beta_i}$ where $u_i$ is a degree $d-m+i$ form
such that $u_i(0,y)=y^{d-m+i}$.
\\
Then we set
$$f_{i+1}=x^{m-i-1}y\left(u_i+\lambda_ix^{\alpha_i-1-m+i}y^{\beta_i}\right)+\lambda_{i+1}x^{\alpha_{i+1}-1}y^{\beta_{i+1}}.$$
So, $xf_{i+1}-yf_i=\lambda_{i+1}x^{\alpha_{i+1}}y^{\beta_{i+1}}$ and for $\lambda_{i+1}\neq 0$, we have
$N_{i+1}\in (f_0, \ldots, f_m)$.
Note that for $i=0$, $u_0=y^{d-m}, \lambda_0=0$; for $i=1$, $u_1=y^{d-m+1}$; thus, inductively, we have constructed
$f_0, \ldots, f_m$ such that $N_i\in I=(f_0, \ldots, f_m)$.
\end{proof}

\begin{remark}[Choosing which corner should be kicked off]\label{selected-corner-kick-off}
Given  $r$ indices $i_1, \ldots, i_r$ such that $1\leq i_1< \ldots <i_r\leq m$, in the inductive construction of
$(f_0, \ldots, f_m)$ of Lemma \ref{corner-kick-off-lemma}, if we let $\lambda_{i_l}=0$ ($1\leq l\leq r$), then none of the monomials
$N_{i_l}$ will be kicked off. So, if $\lambda_{i_l}=0$ for $1\leq l\leq r$, then $N_{i_l}\notin (f_0, \ldots, f_m)$.
\end{remark}

\begin{lemma}[Counting the corner-monomials of $P$]\label{countcornerlem}
Let $T=(1,2, \ldots , d,t_d=t,0)$, $t>0$ and  set $s=d+1-t$. Suppose that $P$ is a partition of diagonal lengths $T$ and difference-one hook code $\mathfrak{Q}(P)=(h_1^{l_1}, \dots, h_n^{l_n})$ (where $s\geq h_1> h_2 >\cdots >h_n \geq 0$). Then the minimum number of generators $b_1(E_P)$ of the monomial ideal $E_P$ is given by the following formula.
$$
\begin{array}{ll}
b_1(E_P)=s+t-n,&\mbox{ if } h_1<s \mbox{ and } h_n>0\\
b_1(E_P)=s+t-n+1, &\mbox{ if } h_1=s \mbox{ and } h_n>0, \mbox{ or } h_1<s \mbox{ and } h_n=0 \\
b_1(E_P)=s+t-n+2,&\mbox{ if } h_1=s \mbox{ and } h_n=0.
\end{array}
$$
\end{lemma}

\begin{proof}
This is an easy count that we obtain by looking at the Ferrers diagram of $P$.
\end{proof}\par
Note that $b_1(E_P)-\kappa(P)$ counts the number of degree $d+1$ corner-monomials we have been able to kick-off.

\begin{example}\label{972bex}
Suppose $T=(1,2,3,4,5,6,7,8,4)$. Then $d=8$, $t_d=4$ and $s=5$. Let $P$ be the partition of diagonal lengths $T$
defined by $P=(9,7^2,6,4^2,2,1)$ (See Figure \ref{fig11}). We have $\mathfrak{Q}(P)=(5,4^2,3)$. The monomial ideal $E_P$ associated to $P$ is
generated by $\left(y^8,xy^7,x^2y^6,x^4y^4,x^7y,x^6y^3,x^9\right)$.
Using Lemma \ref{corner-kick-off-lemma}, we see that the degree $9$ corners of $P$ associated to $x^6y^3$
and {$x^9$} can be kicked-off using the degree $8$ corners associated to the consecutive monomials
{$y^8, xy^7, x^2y^6$}.
\begin{figure}[!h]
\begin{center}
	\begin{tikzpicture}[scale=.8]
	\draw[dashed,very thin] 	(0,0)--(9,0)
						(0,-1)--(7,-1)
						(0,-2)--(7,-2)
						(0,-3)--(6,-3)
						(0,-4)--(4,-4)
						(0,-5)--(4,-5)
						(0,-6)--(2,-6)
						(0,-7)--(1,-7)
						(0,0)--(0,-8)
						(1,0)--(1,-7)
						(2,0)--(2,-6)
						(3,0)--(3,-6)
						(4,0)--(4,-4)
						(5,0)--(5,-4)
						(6,0)--(6,-3)
						(7,0)--(7,-1)
						(8,0)--(8,-1);
	\draw[tips, -{Latex[open,length=8pt,bend]},dashed,red,thick] 	(1.5,-8.5) to [bend left] (6,-3.5);
	\draw[tips, -{Latex[open,length=8pt,bend]},red,thick] (2.5,-7.5) to [bend left] (9.2,-0.5);
	\draw node at (1.5,-8.5) {\faSoccerBallO};
	\draw node at (2.5,-7.5) {\faSoccerBallO};
	\draw[thin, blue,fill=black!5,rounded corners=3mm] (9,0) rectangle (10,-1)
											   (6,-3) rectangle (7,-4)
											   (2,-6) rectangle (3,-7)
											   (1,-7) rectangle (2,-8)
											   (0,-8) rectangle (1,-9);
	\draw[thin, blue,fill=green!10,rounded corners=3mm] (2,-7) rectangle (3,-8)
											      (1,-8) rectangle (2,-9);
	\draw[tips, -{Latex[open,length=8pt,bend]},dashed,red,thick] 	(1.5,-8.5) to [bend left] (6,-3.5);
	\draw[tips, -{Latex[open,length=8pt,bend]},red,thick] (2.5,-7.5) to [bend left] (9.2,-0.5);
	\draw node at (1.5,-8.5) {\faSoccerBallO};
	\draw node at (2.5,-7.5) {\faSoccerBallO};
	\draw[blue] node at (9.5,-0.5) {\footnotesize{$x^9$}}
			    node at (6.5,-3.5) {\footnotesize{$x^6y^3$}}
			    node at (2.5,-6.5) {\footnotesize{$x^2y^6$}}
			    node at (1.5,-7.5) {\footnotesize{$xy^7$}}
			    node at (0.5,-8.5) {\footnotesize{$y^8$}};
	\draw[very thick] (9,0)--(9,-1)--(7,-1)--(7,-3)--(6,-3)--(6,-4)--(4,-4)--(4,-6)--(2,-6)--(2,-7)--(1,-7)--(1,-8)--(0,-8);
         \end{tikzpicture}
         \caption{Kicking off corners of the partition $P=(9,7^2,6,4^2,2,1)$ (Example \ref{972bex}).}\label{fig11}
         \end{center}
	\end{figure}
\end{example}

\medskip
Let $P$ be a partition of diagonal lengths $T$. Suppose $P=(p_1, \ldots , p_m)$, with $p_1\geq p_2 \geq \cdots \geq p_m$. Let
$P^{\prime}=(p_1^{\prime}, \ldots , p_m^{\prime})$ with $p_i^{\prime}=p_i-1$. Let $T^{\prime}$ be the Hilbert function associated to $P^{\prime}$.
If $I$ is an element of the cell $\mathbb{V}(E_P)$ of $\G_T$, then $(I:x)$ is an element of the cell
$\mathbb{V}(E_{P^{\prime}})$ of $\G_{T^{\prime}}$.
In fact we have a morphism $\varphi : \mathbb{V}(E_P) \rightarrow \mathbb{V}(E_{P^{\prime}})$ defined by $I\mapsto (I:x)$ whose fiber is an affine
space of dimension the number of difference-one hooks having their feet at $y^{m-1}$ (\cite[Proposition 2.6]{jy-1}).
\begin{lemma}\label{inductionlem}
Assume that $T=(1, \dots, d, t_d,0)$, and that $P$ is a partition having diagonal lengths $T$ and difference-one hook code $\mathfrak{Q}(P)=(h_1^{l_1}, \dots, h_n^{l_n})$. Set $s=d+1-t_d$. Suppose that $I$ is a generic ideal in the cell $\mathbb V(E_P)$ and let $\bar{I}=(I:x)$.

\begin{itemize}
\item[(a)] If $h_n=0$ then $\bar{I}\in \mathbb V(E_{\bar{P}})$ where $\bar{P}$ is the partition of diagonal lengths $\bar{T}=(1, \dots, d-1, t-1)$ and hook code $\mathfrak{Q}(\bar{P})=(h_1^{l_1}, \dots, h_n^{l_n-1}).$ Furthermore, in this case $\kappa(P)=\kappa(\bar{P})+1.$
\medskip

\item[(b)] If $h_n>0$ then $\bar{I}\in \mathbb V(E_{\bar{P}})$ where $\bar{P}$ is the partition of diagonal lengths $\bar{T}=(1, \dots, d-1, t)$ and  hook code $\mathfrak{Q}(\bar{P})=\big((h_1-1)^{l_1}, \dots, (h_n-1)^{l_n}\big).$ Furthermore, in this case
$$\kappa(P)=\left\{\begin{array}{ll} \kappa(\bar{P}),&\mbox{ if }\kappa(\bar{P})\geq s \\ s,&\mbox{ if } \kappa(\bar{P})=s-1.\end{array}\right.$$

\end{itemize}
\end{lemma}

\par

\begin{proof}

We note that the Ferrers diagram of $\bar{P}$ is always obtained from the Ferrers diagram of $P$ by removing the first column. Let $\mathfrak{\bar{B}}=\{f_1, \dots, f_\kappa\}$ be a minimal set of generators for $\bar{I}$ (the $f_i$'s are ordered according to their leading monomials, from top to bottom in the Ferrers diagram).
\medskip

\noindent Part (a) If $h_n=0$, then $\mathfrak{B}=\{xf_1, \dots, xf_\kappa, y^{d+1}\}$ is a minimal set of generators for $I$. Thus the equality in part (a) holds.
\medskip

\noindent Part (b). If $h_n>0$ then by definition $\bar{s}=s-1$. This in particular implies that in this case $\kappa(\bar{P})\geq s-1$.

Assume that $h_n=1$. Then the leading term of
$f_\kappa$ is $y^d$ (we can even set $f_\kappa=y^d$ here). Let $g$ be a generic polynomial with leading term $y^d$. Then
$\mathfrak{B}=\{xf_1, \dots, xf_{\kappa-1}, g\}$ is a minimal set of generators for $I$. Thus in this case $\kappa(P)=\kappa(\bar{P})$. We also note that since $\kappa(P)\geq s$ (Definition \ref{order_of_Tdef} and Lemma~\ref{kappaTlem})
the equality $\kappa(P)=\kappa(\bar{P})$ in particular implies that when $h_n=1$, we have $\kappa(\bar{P})\geq s$.

\medskip

Next,  assume that $h_n>1$. Suppose $\kappa(\bar{P})=s-1$ (this is the minimum value posssible for $\kappa(\bar{P})$). In this case, all the degree $d$ corner-monomials of $P^{\prime}$ have been kicked-off. After multiplication by $x$, these degree $d$ corner-monomials of $P^{\prime}$ become degree $d+1$
corner-monomials of $P$, so are kicked-off by $(xf_1, \dots, xf_\kappa)$ and therefore $\mathfrak{B}=\{xf_1, \dots, xf_\kappa, g\}$, where $g$ is a generic polynomial with leading term $y^d$, is a minimal set of generators of the generic element of $\mathbb{V}(E_P)$, so
$\kappa(P)=\kappa(\bar{P})+1$. \\
Now, suppose $\kappa(\bar{P})>s-1$. This means that there is at least one  degree-$d$ form in any minimal set of generators of a generic element of
$\mathbb{V}(E_{P^{\prime}})$. So we have $\mathfrak{\bar{B}}=\{f_1, \dots, f_{\kappa-s+1}, f_{\kappa-s+2}, \ldots, f_\kappa\}$,
$\deg( f_{\kappa-s+1})=d$, $\deg( f_{\kappa-s+2})=\cdots = \deg(f_\kappa)=d-1$. Note that $f_\kappa$ has leading monomial $y^{d-1}$. Now,
let $g=yf_\kappa+\lambda f_{\kappa-s+1}$ ($\lambda\neq 0$). Then $xg-y(xf_\kappa)=\lambda xf_{\kappa-s+1}$. Since $\lambda\neq 0$, this
means that $xf_{\kappa-s+1}$ can be kicked off. If $\mathfrak{\bar{B}}=\{f_1, \dots, f_{\kappa-s+1}, f_{\kappa-s+2}, \ldots, f_\kappa\}$
is a minimal set of generators of $(I:x)=\bar{I}$, then $\mathfrak{B}=\{xf_1, \dots, xf_{\kappa-s}, xf_{\kappa-s+2}, \ldots, xf_\kappa,g\}$
is a minimal set of generators of $I$. Thus $\kappa(P)=\kappa(\bar{P})$.
\end{proof}

Recall that for a partition $P$ of diagonal lengths $T$, we denote the minimum number of generators for a generic element $I$ in the cell $\mathbb{V}(E_P)$ by $\kappa(P)$. Also note that, as discussed at the beginning of this section, if $T=(1, \dots, d, t_d,0)$ is a single-block Hilbert function, then a minimal system of generators for $I$ consists of $s=d+1-t_d$ generators of degree $d$ and $(\kappa(P)-s)$ generators of degree $d+1$. The following theorem provides an explicit formula for $\kappa(P)$ in the single block case.

\begin{theorem}[The invariant $\kappa(P)$ for a single block $T$]\label{kappathm}
Assume that $T=(1, \dots, d, t_d=t,0)$ and let $P$ be a partition of diagonal lengths $T$ and difference-one hook code $\mathfrak{Q}(P)=(h_1^{l_1}, \dots, h_n^{l_n})$. For $k=1, \dots, n$, let $\tau_k=\sum\limits_{i=k}^nl_i-h_k.$ Then
\begin{equation}\label{kappaformula}\kappa(P)=s+\max\{t+1-s, 0, \tau_k\}_{k=1, \dots, n}.\end{equation}

\end{theorem}

\begin{proof}
We prove the theorem by induction on $d$.
\bigskip

First assume that $d=2$.
\medskip

If $t=2$, then $s=1$. In this case there are three partitions of diagonal lengths $T$, namely
\begin{itemize}
\item[(i)] Partition $P=(3,2)$ with hook code $\mathfrak{Q}(P)=(1^2)$, $\kappa(P)=3$ and $$s+\max\{t+1-s,0, \tau_1\}=1+\max\{2,0, 2-1\}=3;$$

\item[(ii)] Partition $P=(3,1^2)$ with hook code $\mathfrak{Q}(P)=(1,0)$, $\kappa(P)=3$ and $$s+\max\{t+1-s,0, \tau_1, \tau_2\}=1+\max\{2,0, 2-1, 1-0\}=3;$$

\item[(iii)] Partition $P=(2^2,1)$ with hook code $\mathfrak{Q}(P)=(0^2)$, $\kappa(P)=3$ and $$s+\max\{t+1-s,0, \tau_1\}=1+\max\{2,0,2-0\}=3.$$
\end{itemize}
\medskip

On the other hand, if $t=1$, then $s=2$. In this case, there are three partitions of diagonal lengths $T$, namely

\begin{itemize}
\item[(i)] Partition $P=(3,1)$ with hook code $\mathfrak{Q}(P)=(2)$, $\kappa(P)=2$ and $$s+\max\{t+1-s,s, s+\tau_1\}=2+\max\{0,0,1-2\}=2;$$

\item[(ii)] Partition $P=(2^2)$ with hook code $\mathfrak{Q}(P)=(1)$, $\kappa(P)=3$ and $$s+\max\{t+1-s,0, \tau_1\}=2+\max\{0,0,1-1\}=2;$$

\item[(iii)] Partition $P=(2,1^2)$ with hook code $\mathfrak{Q}(P)=(0)$, $\kappa(P)=3$ and $$s+\max\{t+1-s,0, \tau_1\}=2+\max\{0,0,1-0\}=3.$$
\end{itemize}

This shows that the desired equality holds when $d=2$.
\bigskip

Now assume that $d>2$ and that Equation (\ref{kappaformula}) holds for any partition of diagonal lengths $(1, \dots, d',t,0)$ with $d'<d$.

Suppose that $P$ is a partition of diagonal lengths $T=(1, \dots, d, t)$ and hook code  $\mathfrak{Q}(P)=(h_1^{l_1}, \dots, h_n^{l_n})$. Let $\bar{P}$ be the partition associated to $P$ defined in Lemma \ref{inductionlem}. Then by the inductive hypothesis Equation \ref{kappaformula} holds for $\kappa(\bar{P})$.

\medskip

{\bf Case 1.} Assume that $h_n=0$. Then $\bar{t}=t-1$, $\bar{s}=s$, and for $k=1, \dots, n$, we have $\bar{\tau}_k=\tau_k-1$. Thus
$$\begin{array}{ll}
\kappa(\bar{P})&=\bar{s}+\max\{\bar{t}+1-\bar{s}, 0, \bar{\tau}_k\}_{k=1, \dots, n}\\
&=s+\max\{t-s,0, \tau_k-1\}_{k=1, \dots, n}.
\end{array}$$

Since $\tau_n=l_n-h_n=l_n\geq1$, we have

$$\begin{array}{ll}
\max\{t-s,0, \tau_k-1\}_{k=1, \dots, n}&=\max\{t-s,\tau_k-1\}_{k=1, \dots, n}\\ \\
&=\max\{t+1-s, \tau_k\}_{k=1, \dots, n}-1.
\end{array}$$

Thus using part (a) of Lemma \ref{inductionlem} we have

$$\begin{array}{ll}
\kappa(P)&=\kappa(\bar{P})+1\\
&s+\max\{t+1-s, \tau_k\}_{k=1, \dots, n}\\
&s+\max\{t+1-s,0, \tau_k\}_{k=1, \dots, n}.
\end{array}$$

\medskip

{\bf Case 2.} Assume that $h_n>0$. Then $\bar{t}=t$, $\bar{s}=s-1$, and for $k=1, \dots, n$, we have $\bar{\tau}_k=\tau_k+1$. By the inductive hypothesis

$$\begin{array}{ll}
\kappa(\bar{P})&=\bar{s}+\max\{\bar{t}+1-\bar{s}, 0, \bar{\tau}_k\}_{k=1, \dots, n}\\
&=s-1+\max\{t+1-s+1,0, \tau_k+1\}_{k=1, \dots, n}.
\end{array}$$

If $h_n=1$, then $\tau_n=l_n-1\geq 0$. Furthermore, if $\kappa(\bar{P})\geq s$ then $t+1\geq s$ or $\tau_k\geq 0$ for some $k$. In either of these cases, we have
 $$
\max\{t+1-s+1,0, \tau_k+1\}_{k=1, \dots, n}=\max\{t+1-s+1, \tau_k+1\}_{k=1, \dots, n}.$$ Thus, using Lemma \ref{inductionlem}, we have

$$\begin{array}{ll}
\kappa(P)&=\kappa(\bar{P})\\
&=s-1+\max\{t+1-s+1, \tau_k+1\}_{k=1, \dots, n}\\
&=s+\max\{t+1-s, \tau_k\}_{k=1, \dots, n}\\
&=s+\max\{t+1-s,0, \tau_k\}_{k=1, \dots, n}.
\end{array}$$

Finally, if $\kappa(\bar{P})=s-1$, then $t+1\leq s-1$ and $\tau_k+1\leq 0$, for all $k=1, \dots, n$. This in particular implies that in this case $s+\max\{t+1-s,0, \tau_k\}_{k=1, \dots, n}=s$.

By Lemma \ref{inductionlem}, we also have

$$\begin{array}{ll}
\kappa(P)&=\kappa(\bar{P})+1\\
&=s-1+1\\
&=s
\end{array}$$
\end{proof}\par
Recall that a partition in $\mathcal P(T)$ is \emph{special} if $\kappa(P)>\kappa (T)$ from Equation~\eqref{kappaTeq}.
\begin{corollary}[Special partitions]\label{special1cor}
Assume that $P$ is a single block partition.  Then $P$ is special if and only if
some $\tau_k$ from Theorem \ref{kappathm} satisfies $\tau_k>\delta$ where $\delta=\max\{t_d+1-s,0\}$.
\end{corollary}
\begin{proof} This follows from Equation \ref{kappaTeq} and Theorem \ref{kappathm}.
\end{proof}
\begin{remark}
We note that if at least one entry in the hook code of $P$ is zero, then $\tau_n=l_n>0$. Thus, in this case $\kappa(P)\geq s+1$. This in particular implies that in part (b) of Lemma \ref{inductionlem}, if $h_n=1$ then the hook code of $\bar{P}$ has a zero entry and therefore $\kappa(\bar{P})\geq \bar{s}+1=s$.

\end{remark}

\section{Partitions in $\mathcal P(T)$ having a given number of generators, for single-block $T$.}\label{countSingleSection}
We begin with a result counting the total number of partitions having diagonal lengths a single-block Hilbert function $T$. We then in Theorem \ref{countingthm} count those associated to a given generic number of generators $\kappa(P)$. Throughout the section $T$ will be a single-block Hilbert function $T=(1,2,\ldots, d,t_d=t,0)$ of Equation~\eqref{Tsingleeqn}, we let $s=d+1-t$ and we set $\delta=\max\{t+1-s,0\}=\max\{2t-d,0\}$.

\begin{lemma}\label{count1blockcor}
The number of partitions having the single-block diagonal lengths $T=(1,2,\ldots, d, t_d=t)$ satisfies
\begin{equation}\label{count1blockeq}
\# \mathcal P(T)=\binom{s+t}{t}.
\end{equation}
\end{lemma}
\begin{proof} By Theorem \ref{PTtoQTthm}, $\# \mathcal P(T)$ counts the total number of partitions whose Ferrers diagram
can be placed in a $t\times s$ box $\mathfrak B_d(T)$ or, equivalently, the number of lattice paths from $(0,0)$ to $(s,t)$, which satisfies \eqref{count1blockeq}.
\end{proof}

\par
By Theorem \ref{kappathm}, the number of generators for a generic ideal in the cell $\mathbb V(E_P)$ is $\kappa(P)=s+\max\{\delta, \tau_k\}_{k=1, \dots, n}.$
In particular, for all partitions $P$ of diagonal lengths $T$, we have $$\kappa(T)=s+\delta\leq \kappa(P) \leq s+t.$$

\begin{theorem}[Number of special partitions of diagonal lengths $T$]\label{countingthm}

Let $T=(1, \dots, d, t,0)$, $s=d+1-t$, and $\delta=\max\{t+1-s,0\}$. Assume that $k$ is an integer such that $s+\delta< k \leq s+t$. Then the number of partitions $P$ of diagonal lengths $T$ and $\kappa(P)\geq k$ is
\begin{equation}\label{countsinglegekeq}
\binom{s+t}{k}.
\end{equation}
In particular the number of special partitions of diagonal lengths $T$ is
\begin{equation}\label{numberspecial1eq}
\binom{s+t}{s+\delta+1}=\binom{s+t}{\min\{s-1,t\}}.
\end{equation}
And the number of non-special partitions of diagonal lengths $T$ is
\begin{equation}\label{numbernonspecial1eq}
\binom{s+t}{s}-\binom{s+t}{s+\delta+1}.
\end{equation}
\end{theorem}

\begin{proof}

We use induction on $d$.
For $d=2$ the statement holds by the discussion of this case in the proof of Theorem \ref{kappathm}.

\medskip

Now assume that $d>2$. For a partition $P$ with hook code $\mathfrak{Q}(P)=(h_1^{l_1}, \dots, h_n^{l_n}) $, we define $\bar{P}$ as in Lemma \ref{inductionlem}.

\medskip

{\bf Case 1.} Assume that $P$ is such that $h_n=0$.

Then by Lemma \ref{inductionlem}, we have $\kappa(P)=\kappa(\bar{P})+1$. Thus $\kappa(P)\geq k$ if and only if $\kappa(\bar{P})\geq k-1$.

Moreover, in this case $\bar{s}=s$ and $\bar{t}=t-1$. Hence in this case,
$$\bar{\delta}=\max\{\bar{s}-\bar{t},0\}=\left\{\begin{array}{ll}\delta-1,&\mbox { if } \delta\geq 1\\0,&\mbox { if } \delta=0\end{array}\right.$$

If $\delta\geq 1$, then $\bar{\delta}=\delta-1$ and therefore $\bar{s}+\bar{\delta}<k-1\leq \bar{s}+\bar{t}$ holds. On the other hand, if $\delta=0$ and $s+1<k$, we still have the desired inequalities $\bar{s}<k-1\leq \bar{s}+\bar{t}$. Thus by the inductive hypothesis, in these cases, the number of partitions $P$ with $\kappa(P)\geq k$ satisfies

$$\binom{\bar{s}+\bar{t}}{k-1}=\binom{s+t-1}{k-1}.$$

Finally, if $\delta=0$ and $k=s+1$, then all partitions $\bar{P}$ have at least $k-1=s=\bar{s}$ generators. The number of such partitions satisfies

$$\binom{\bar{s}+\bar{t}}{\bar{t}}=\binom{s+t-1}{t-1}=\binom{s+t-1}{s}=\binom{s+t-1}{k-1}.$$

Thus there are $\binom{s+t-1}{k-1}$ partitions $P$ with $\kappa(P)\geq k$ and a zero entry in the hook code.
\medskip

{\bf Case 2.} Assume that $h_n> 0$.  if $s+\delta<k\leq s+t$ and $\kappa(P)\geq k$, then in particular $\kappa(P)>s$. Thus by part (b) of Lemma \ref{inductionlem}, we have $\kappa(\bar{P})\geq s$, and $\kappa(P)=\kappa(\bar{P})$.

We also have $\bar{s}=s-1$ and $\bar{t}=t$. Thus $\bar{\delta}=\max\{(t+1-s)+1,0\}.$ In other words, $\bar{\delta}=\delta+1$ if $s\leq t+1$, and $\bar{\delta}=\delta=0$, otherwise. This shows that if $s+\delta<k$ then $\bar{s}+\bar{\delta}<s$ as well. We also note that if $k<t+s$, then $k\leq \bar{s}+\bar{t}$, while for $k=s+t$, there is no partition $\bar{P}$ with $\kappa(P)\geq k$.

By the inductive hypothesis, the number of partitions $P$ with $\kappa(P)\geq k$ is $$\binom{\bar{s}+\bar{t}}{k}=\binom{s+t-1}{k}.$$

Combining the count for the two possible cases, we see that the total number of partitions $P$ with $\kappa(P)\geq k$ is
$$\binom{s+t-1}{k-1}+\binom{s+t-1}{k}=\binom{s+t}{k}.$$
The first equality of Equation \eqref{numberspecial1eq} is the special case $k=s+\delta+1$, the second follows from $s+t-(s+\delta+1)=\min\{s-1,t\}$.\par
The count of non-special partitions follows from Lemma \ref{count1blockcor} giving the total number $\binom{s+t}{t}$ of partitions of diagonal lengths $T$ and the count of special partitions.
\end{proof}

\begin{cor}\label{singlecountcor}
Let $T=(1, \dots, d, t_d=t,0)$, $s=d+1-t$, and $\delta=\max\{t+1-s,0\}$. For a positive integer $k$, we define $\mu(T,k)$ to be the number of partitions $P$ with diagonal lengths $T$ and $\kappa(P)=k$. Then
\[
\mu(T,k)=
\left\{
\begin{array}{ll}
    \binom{s+t}{s}-\binom{s+t}{s+\delta+1},   & \mbox{ if } k=s+\delta \mbox{ (non-special $P$)}, \\\\
    \binom{s+t}{k}-\binom{s+t}{k+1},           & \mbox{ if } s+\delta<k\leq s+t,\\\\
    0,                                         & \mbox{ otherwise. }
\end{array}
\right.
\]
\end{cor}
\begin{remark}
Obviously, by its definition, for all $k$, the invariant $\mu(T,k)$ is non-negative. We also note that the inequalities \[s+t\leq 2\, \max\{s,t\}\leq 2\, \max\{s, t+1\}=2(s+\delta)\]
imply that $s+\delta\leq \frac{s+t}{2}$. Thus, if $s+\delta<k\leq s+t$, then $\binom{s+t}{k}>\binom{s+t}{k+1}$.
As for $k=s+\delta$, if $s>t$ then $s>\frac{s+t}{2}$ and therefore \[\binom{s+t}{s}>\binom{s+t}{s+1}=\binom{s+t}{s+\delta+1}.\]

On the other hand, if $s\leq t$, then $s\leq \frac{s+t}{2}$. Therefore, $\binom{s+t}{s-2}<\binom{s+t}{s}$. Moreover, we have $\delta=(t+1)-s$, and therefore,
 \[\binom{s+t}{s+\delta+1}=\binom{s+t}{t+2}=\binom{s+t}{s-2}<\binom{s+t}{s}.\]

 We also note that for $k=s+\delta$,  the number $\mu(T,k)$ is the coefficient of the degree $k=s+\delta$ term in $(1+z)^{s+t}\left(z^\delta-\frac{1}{z}\right)$ while
for $s+\delta<k< s+t$, the number $\mu(T,k)$ is the same as the coefficient of the degree $k$ term in $(1+z)^{s+t}\left(1-\frac{1}{z}\right)$.
\end{remark}

\begin{remark}
In \cite[Section 2C]{IY} it was shown that for any single-block Hilbert function $T$, there will be a unique minimal finite set of special partitions of diagonal lengths $T$, such that any special partition is in the closure of the minimal set. The following example shows that the special cells do not form an irreducible subfamily of $G_T$.
\end{remark}
\begin{example}[Single block table]\label{table2ex}
Let $T=(1,2,3,4,2,0)$, then $t=2,s=3$ and  $\mathfrak B(T)=({\mathfrak B}_4(T))=((2\times 3)),$ and there are $\binom{5}{2}=10$ partitions of diagonal lengths $T$.  We give Table \ref{420table} for these, specifying the hook code, and $\kappa(P)$ for each. Here $\delta(T)=\max\{0,s+1-t\}=0$ and $ \kappa(T)=3$.  We have placed
conjugate partitions in symmetric positions from the center line; the two middle partitions of hook codes $(3,0)$ and $(2,1)$ are self-conjugate. Note also that the conjugate partition $P^\vee$ has the complementary hook code in ${\mathfrak B}_4(T)=(3,3)$. \par
Figure~\ref{specializefig} gives the specialization diagram for  $\mathcal P(T)$, corresponding to inclusion of the Ferrers diagrams for the hook codes $\mathfrak h_4(P)$ (on the left).  \par
We see from the table that the cells in
$\kappa(P)\ge 4$ are the union of the closures of cells having hook codes  $ (1,1)$ and $(3,0)$: $ \kappa(P) = 4$ includes also the cells with hook codes $(2,0)$ and $ (1,0),$ while the cell with hook code $(0,0)$ is the unique with $\kappa(P)=5$ (these cells are colored red/blue on the left of Figure~\ref{specializefig}.)  Thus, the subvariety of cells corresponding to special partitions is here the union of two irreducible components, of dimensions three (closure of $(3,0)$) and two (closure of $(1,1)$), respectively.  \par

\end{example}
\begin{table}
\begin{center}$
\begin{array}{|c|c|c|c|c|c|}
\hline
P&\mathfrak h_4&\tau_1&\tau_2&\kappa(P)\\
\hline\hline
(5,4,2,1)&(3,3)&-1&&3\\
\hline
(5,3,3,1)&(3,2)&-1&-1&3\\
\hline
( 5,3,2,2)&(3,1)&0&-1&3\\
\hline
(4,4,3,1)&(2,2)&0&&3\\
\hline\hline
(5,3,2,1,1)&(3,0)&-2&1&4\\
\hline
(4,4,2,2)&(2,1)&0&0&3\\
\hline\hline
(4,3,3,2)&(1,1)&1&0&4\\
\hline
(4,4,2,1,1)&(2,0)&0&1&4\\
\hline
(4,3,3,1,1)&(1,0)&1&1&4\\
\hline
(4,3,2,2,1)&(0,0)&2&&5\\
\hline
\end{array}$
\end{center}
\caption{Table of $\mathcal P(T), T=(1,2,3,4,2,0)$. See Example \ref{table2ex}.}\label{420table}
\end{table}

\footnotesize
\begin{figure}
		$\xymatrix{&\underline{\phantom{a}\mathfrak h_4(P)\phantom{a}}&\\&{(3,3)}\ar[d]& \\
		&{(3,2)}\ar[ld]\ar[rd]&\\
		(3,1)\ar[d] \ar[rrd]&&(2,2)\ar[d]\\
		{\textcolor{red}{(3,0)}}\ar[d]&&(2,1)\ar[d] \ar[lld]\\
		{\textcolor{red}{(2,0)}}\ar[rd]&&{\textcolor{red}{(1,1)}}\ar[ld]\\
		&{\textcolor{red}{(1,0)}}\ar[d]&\\
		&{\textcolor{blue}{(0,0)}}&}$
			\qquad\quad $\xymatrix{&\underline{\phantom{a}P\phantom{a}}&\\&{(5,4,2,1)}\ar[d]& \\
		&{(5,3,3,1)}\ar[ld]\ar[rd]&\\
		(5,3,2,2)\ar[d] \ar[rrd]&&(4,4,3,1)\ar[d]\\
		(5,3,2,1,1)\ar[d]&&(4,4,2,2)\ar[d] \ar[lld]\\
		(4,4,2,1,1)\ar[rd]&&(4,3,3,2)\ar[ld]\\
		&(4,3,3,1,1)\ar[d]&\\
		&(4,3,2,2,1)&}$
				\caption{Specialization diagram for $\mathcal P(T), T=(1,2,3,4,2,0)$. See Example \ref{table2ex}.}\label{specializefig}
			\end{figure}
\normalsize
\newpage
\section{Number of generators for multiblock partitions.}\label{numGensMultiSection}
Throughout this section, $T=(1, \dots, d, t_d, \dots, t_{\sf j}, 0)$, and $P$ is a partition  lengths $T$ and difference-one hook code $\mathfrak{Q}(P)=(\mathfrak{h}_d, \dots , \mathfrak{h}_{\sf j})$.

Recall from Equation \eqref{HFdecomposition} that for $i=d, \dots, {\sf j}$, we set $\delta_i=t_{i-1}-t_i$ and $$T_i=(1, \dots, \delta_i+\delta_{i+1}, \delta_{i+1}, 0).$$
As we saw in Definition \ref{Pidef1} and Lemma \ref{hookP(i)lem}, a partition $P\in \mathcal P(T)$ can be decomposed into single-block ``component'' partitions $P_i$. For $i=d, \dots, {\sf j}$, the partition $P_i$ has diagonal lengths $T_i$ and difference-one hook code $\mathfrak{h}_i$. We note that, although by construction the hook code for $P_i$ is $\mathfrak{h}_i$, the corresponding hook degree in $P_i$ is $\delta_i+\delta_{i+1}$ and not $i$. We showed in Theorem~\ref{projectionthm} that the cells $\mathbb V(E_P)$ are naturally the product of the corresponding cells  $\mathbb V(E_{P_i})$.
\par
In this section we count the minimum number of generators for ideals in the cell associated to an arbitrary partition. The main results of this section are the following Theorem \ref{componentTheorem} relating to the components, and Theorem~\ref{extra_gens_in_given_degree}, which specifies the number $\beta_{i,0}(P)$ of degree-$i$ generators of an ideal $I$ defining a generic element of $\mathbb V(E_P)$.

\begin{theorem}[Decomposition of $\kappa(P)$ into components]\label{componentTheorem}
Let $P$ be a partition of lengths $T=\left(1,2,\dots ,d,t_d,\dots ,t_{\sf j},0\right)$.  Then
\begin{equation}\label{kappaDeompositionEq}
\kappa(P) = \sum\limits_{i=d}^{\sf j}\kappa(P_i)-(t_d-t_{\sf j})-({\sf j}-d).
\end{equation}
Moreover, a minimal set of generators for a generic ideal in the cell $\mathbb{V}(E_P)$ includes $d+1-t_d$ generators of degree $d$ and  $\kappa(P_i)-(\delta_i+1)$ generators of degree $i+1$, for $i\in [d, {\sf j}]$.
\end{theorem}
The value $\kappa(P_i)$ for a single block partition was determined in Theorem~\ref{kappathm}.
We establish the Theorem by proving the series of statements that follow, culminating in Corollary \ref{multicountcor}, which is an avatar of the Theorem.

The $i$-th block $\mathfrak{h}_i(P)$ of the hook code $\mathfrak{Q}(P)$ is a key element of our statements. So we first give a formula for $\kappa(P_i)$ when $\mathfrak{h}_i(P)$ is empty.  Recall that the case $\mathfrak{h}_i=\emptyset$ occurs when $t_i=t_{i+1}$ and in this case, we can think of $P_i$ as the basic triangle $\Delta_{\delta_i}$ (Remark \ref{empty_hook_code_block_partition}).
\begin{proposition}\label{kappa-of-empty-bloc}
If the block $\mathfrak{h}_i(P)$ is empty, then $\kappa(P_i)=\delta_i+1$.
\end{proposition}
\begin{proof}
Recall that the case $\mathfrak{h}_i=\emptyset$ occurs when $t_i=t_{i+1}$ and in this case, we can think of $P_i$ as the basic triangle
$\Delta_{\delta_i}$ (see Remark \ref{empty_hook_code_block_partition}). It is then clear that the monomial ideal whose cobasis is the basic triangle
$\Delta_{\delta_i}$ has exactly $\delta_i+1$ generators, the degree $\delta_i$ monomials. Note that the basic triangle
$\Delta_{\delta_i}$ is empty when $\delta_i=0$.
\end{proof}
\begin{example}\label{kappa-of-empty-bloc-example}
Consider the partition $P=(8,7,4,2,1,1)$ of diagonal lengths $T=(1,2,3,4,5,4,2,2,0)$. The degree 6 block $\mathfrak{h}_6(P)$ is empty ($t_6=t_7=2$).
$\kappa(P_6)=\delta_6+1=3$ and this number corresponds to the three border monomials $(x^4y^2,x^3y^3,y^6)$ of $E_P$.
\end{example}

From now on, we will assume that the $i$-th block $\mathfrak{h}_i(P)$ of the hook code $\mathfrak{Q}(P)$ is not empty.
For $i=d, \dots, {\sf j}$, the degree $i$ block $\mathfrak{h}_i$ in the hook code of $P$ can be written as
\begin{equation}\label{hookcomponent}
\mathfrak{h}_i=\left(h_{i,1}^{l_{i,1}}, \dots , h_{i,k}^{l_{i,k}}, \dots, h_{i,n_{i}}^{l_{i,n_{i}}}\right)
\end{equation}

where $\sum\limits_{k=1}^{n_i}l_{i,k}=\delta_{i+1}$ and
$
\delta_i+1\geq h_{i,1}>h_{i,2}>\cdots >h_{i,n_i}\geq 0.
$
\par

As in the single-block case, we construct two sequences of integers $R(P)_i$ and $G(P)_i$ that allow us to count degree $i+1$ relations and degree $i+1$ generators of the monomial ideal $E_P$ associated to $P$.\\

 \begin{definition}[Sequences of degree $i+1$ relations and generators]\label{degree_i+1_rel_gen_sequences}  \
 Denote by $R_{i,k}$ the set of degree $i+1$ relations below the $\left(l_{i,1}+ \dots + l_{i,k}\right)$-th degree $i$ hand of $P$.
 Then we let $R(P)_i=\left(r_{i,1}, \dots , r_{i,n_{i}} \right)$ where $r_{i,k}$ counts the number of  elements of  $R_{i,k}$.

\medskip
\noindent
Also, let $G_{i,1}$ be the set of of degree $i+1$ corner-monomials of $E_P$ and
 for $2\leq k\leq n_i$, denote by $G_{i,k}$ the set of degree $i+1$ corner-monomials of $E_P$ below the
 $\left(l_{i,1}+ \dots + l_{i,k-1}+1\right)$-th degree $i$ hand of $P$. Then $G(P)_i=\left(g_{i,1}, \dots , g_{i,n_{i}} \right)$
 where $g_{i,k}$ counts the number of  elements of  $G_{i,k}$.
 \end{definition}

 \begin{remark}[chains of degree $i+1$ relations and generators]\label{chains_of_degree_i+1_rel_gen}
 Note that using the above notation, we have  the following sequences of inclusions.
 $$
 \begin{array}{l}
 R_{i,n_i}\subset R_{i,n_i-1} \subset \cdots \subset R_{i,k+1}\subset R_{i,k} \subset \cdots \subset R_{i,1}\\
 G_{i,n_i}\subset G_{i,n_i-1} \subset \cdots \subset G_{i,k+1}\subset G_{i,k} \subset \cdots \subset G_{i,1}
 .
 \end{array}
 $$
 So by definition, $R(P)_i=\left(r_{i,1}, \dots , r_{i,n_{i}} \right)$ and $G(P)_i=\left(g_{i,1}, \dots , g_{i,n_{i}} \right)$ are non-increasing sequences.

\end{remark}

 \bigskip
 \noindent
 We now use the hook code $\mathfrak{h}_i=\left(h_{i,1}^{l_{i,1}}, \dots,  h_{i,k}^{l_{i,k}}, \dots , h_{i,n_{i}}^{l_{i,n_{i}}}\right)$ to give formulas for the integers $r_{i,k}$ and $g_{i,k}$ of Definition \ref{degree_i+1_rel_gen_sequences}.

 \begin{observation}\label{degree_i+1_rel_gen_observation}
 By definition of the difference-one hook code, for any integer $1\leq k\leq n_i$, there are $(h_{i,k}-h_{i,k+1})$
 \textsl{horizontal-border monomials} below the $\left(l_{i,1}+ \dots + l_{i,k}\right)$-th degree $i$ hand and above the
 $\left(l_{i,1}+ \dots + l_{i,k}+1\right)$-th degree $i$ hand of $P$.
 Let $\left(M_{i,1}, \dots ,M_{i,h_{i,k}-h_{i,k+1}}\right)$ be the list of these monomials, ordered from top to bottom.
 Then there is exactly one degree $(i+1)$ relation between any two consecutive monomials $M_{i,m}$ and $M_{i,m+1}$
 ($1\leq m\leq h_{i,k}-h_{i,k+1}$). So, we have exactly $(h_{i,k}-h_{i,k+1}-1)$ degree $i+1$ relations below the $\left(l_{i,1}+ \dots + l_{i,k}\right)$-th degree-$i$ hand and above the $\left(l_{i,1}+ \dots + l_{i,k}+1\right)$-th degree-$i$ hand of $P$. \\
 Also we observe that each degree-$i$ hand of $P$ is just left of a degree $(i+1)$ \textsl{vertical-border monomial} of $E_P$.
 We use this property to count the degree $(i+1)$ corner monomials of $E_P$.
 \end{observation}
\begin{lemma}[Counting degree $i+1$ relations and corner monomials]\label{degree_i+1_rel_gen_formulas}
Let $\mathfrak{h}_i=\left(h_{i,1}^{l_{i,1}}, \dots,  h_{i,k}^{l_{i,k}}, \dots , h_{i,n_{i}}^{l_{i,n_{i}}}\right)$ be the $i$-th block of the hook code $\mathfrak{Q}(P)$. \\
 Then the sequences $R(P)_i=\left(r_{i,1}, \dots , r_{i,n_{i}} \right)$ and $G(P)_i=\left(g_{i,1}, \dots , g_{i,n_{i}} \right)$ of Definition \ref{degree_i+1_rel_gen_sequences} are given by the following numbers.
 \begin{enumerate}[(a)]
 \item If $h_{i,n_i}>0$, then
 \begin{itemize}
 \item[$\bullet$]
 $
 g_{i,1}=\left\{
 \begin{array}{lr}
 \delta_{i+1}-(n_i-1) & \text{if} \ h_{i,1}=\delta_i +1\\
 \delta_{i+1}-n_i & \text{if} \ h_{i,1}<\delta_i +1
 \end{array}
 \right.
 $
 \item[$\bullet$]
 $
 \displaystyle{
 g_{i,k}=\sum_{j=k}^{j=n_i}l_{i,j}-(n_i+1-k)
 }
 $, for $2\leq k\leq n_i$
 \item[$\bullet$]
 $
 r_{i,k}=h_{i,k}-(n_i+1-k)
 $,
 for $1\leq k\leq n_i$.
 \end{itemize}
 \item If $h_{i,n_i}=0$, then
 \begin{itemize}
 \item[$\bullet$]
 $
 g_{i,1}=\left\{
 \begin{array}{lr}
 \delta_{i+1}-(n_i-2) & \text{if} \ h_{i,1}=\delta_i +1\\
 \delta_{i+1}-(n_i-1) & \text{if} \ h_{i,1}<\delta_i +1
 \end{array}
 \right.
 $
 \item[$\bullet$]
 $
 \displaystyle{
 g_{i,k}=\sum_{j=k}^{j=n_i}l_{i,j}-(n_i-k)
 }
 $, for $2\leq k\leq n_i$
 \item[$\bullet$]
 $
 r_{i,k}=h_{i,k}-(n_i-k)
 $
 for $1\leq k\leq n_i$.
 \end{itemize}
\end{enumerate}
\end{lemma}

\begin{proof}
Note that by definition of the difference-one hook code, we have
$\delta_i+1\geq h_{i,1}> \dots > h_{i,n_{i}}\geq 0$  and $\displaystyle{\sum_{k=1}^{k=n_i}l_{i,k}=\delta_{i+1}}$. \\
The numbers given in the Lemma come directly from Observation \ref{degree_i+1_rel_gen_observation}.
\begin{enumerate}[(a)]
\item If $h_{i,n_i}>0$, then we have
\begin{itemize}
\item[$\bullet$]
$\displaystyle{g_{i,1}=c_{i,1}+(l_{i,2}-1)+\cdots + (l_{i,n_i}-1)=c_{i,1}+\sum_{k=2}^{k=n_i}l_{i,k}-(n_i-1)}$ where \\
 $
 \displaystyle{
 c_{i,1}=\left\{
 \begin{array}{ll}
 l_{i,1} & \text{if} \ h_{i,1}=\delta_i+1 \\
 l_{i,1}-1 & \text{if} \ h_{i,1}<\delta_i+1
 \end{array}
 \right.
 }
 ,
 $
and for $2\leq k\leq n_i$,  we get \\
$\displaystyle{g_{i,k}=(l_{i,k}-1)+\cdots + (l_{i,n_i}-1)=\sum_{j=k}^{j=n_i}l_{i,j}-(n_i+1-k)}$.
\item[$\bullet$]
$r_{i,k}=(h_{i,k}-h_{i,k+1}-1) + \cdots + (h_{i,n_i-1}-h_{i,n_i}-1)+(h_{i,n_i}-1)=h_{i,k}-\left(n_i -(k-1)\right)$
\end{itemize}
\item If $h_{i,n_i}=0$, then we have
\begin{itemize}
\item[$\bullet$]
$\displaystyle{g_{i,1}=c_{i,1}+(l_{i,2}-1)+\cdots +(l_{i,n_{i-1}}-1)+l_{i,n_i}=c_{i,1}+\sum_{k=2}^{k=n_i}l_{i,k}-(n_i-2)}$ where \\
 $
 \displaystyle{
 c_{i,1}=\left\{
 \begin{array}{ll}
 l_{i,1} & \text{if} \ h_{i,1}=\delta_i+1 \\
 l_{i,1}-1 & \text{if} \ h_{i,1}<\delta_i+1
 \end{array}
 \right.
 }
 ,
 $
and for $2\leq k\leq n_i$,  we get \\
$\displaystyle{g_{i,k}=(l_{i,k}-1)+\cdots + (l_{i,n_{i-1}}-1)+l_{i,n_i}=\sum_{j=k}^{j=n_i}l_{i,j}-(n_i-k)}$.
\item[$\bullet$]
$r_{i,k}=(h_{i,k}-h_{i,k+1}-1) + \cdots +(h_{i,n_{i-1}}-1)=h_{i,k}-\left(n_i -k\right)$.
\end{itemize}
\end{enumerate}
\end{proof}

\begin{remark}
Using the formulas for $r_{i,k}$ and $g_{i,k}$, if we let $\theta_{i,k}=g_{i,k}-r_{i,k}$ ($1\leq k\leq n_i$),
we obtain
\begin{itemize}
\item[$\bullet$]
$
\theta_{i,1}=\left\{
\begin{array}{ll}
\delta_{i+1}+1-h_{i,1} & \text{if} \ h_{i,1}=\delta_i+1\\
\delta_{i+1}-h_{i,1} & \text{if} \ h_{i,1}<\delta_i+1
\end{array}
\right.
$
\item[$\bullet$]
$
\displaystyle{
\theta_{i,k}=\sum_{j=k}^{j=n_i}l_{i,j}-h_{i,k}
}
$ for $2\leq k\leq n_i$.
\end{itemize}
\end{remark}

\begin{remark}\label{compact_formulas_for_gik_rik}
For $1\leq k\leq n_i$, the formulas in Lemma \ref{degree_i+1_rel_gen_formulas}  can be rewritten in a compact way using the invariants of the hook code Equation \eqref{hookcomponent}
\begin{itemize}
\item[$\bullet$]
$
 \displaystyle{
 g_{i,k}=\sum_{j=k}^{j=n_i}l_{i,j}-(n_i+1-k)+\max\{1-h_{i,n_i},0\}+\max\{h_{i,k}-\delta_i,0\}
 }
 $;
 \item[$\bullet$]
 $
 r_{i,k}=h_{i,k}-(n_i+1-k) +\max\{1-h_{i,n_i},0\}
 $.
\end{itemize}
So,  $
\displaystyle{
\theta_{i,k}=\sum_{j=k}^{j=n_i}l_{i,j}-h_{i,k}+\max\{h_{i,k}-\delta_i,0\}
}
$ for $1\leq k\leq n_i$. These are the ingredients for the formula Equation \eqref{beta0eq} for $\beta_{i+1,0}(P)=N_{i+1}-N_i$ in Theorem \ref{extra_gens_in_given_degree} for the number of generators of degree-$i$ for an ideal $ I$ defining a generic element $A $ of $\mathbb V(E_P)$: we will show there that $\beta_{i+1,0}(P)$ is just the integer $\max\left\{\delta_{i+1}-\delta_i,0,\tau_{i,k}\right\}_{1\leq k\leq n_i}$ of the next lemma.
\end{remark}

\begin{lemma}
With the above notation, for $1\leq k\leq n_i$, let
$
\displaystyle{
\tau_{i,k}=\sum_{j=k}^{j=n_i}l_{i,j}-h_{i,k}
}
$.
Then
$
\displaystyle{
\max\left\{0,\theta_{i,k} \right\}_{1\leq k\leq n_i}=\max\left\{\delta_{i+1}-\delta_i,0,\tau_{i,k}\right\}_{1\leq k\leq n_i}
}
$.
\end{lemma}
\begin{proof}[Proof] If $h_{i,1}=\delta_{i}+1$, then $\theta_{i,1}=\delta_{i+1}-\delta_i=\tau_{i,1}+1$.
\\
We then have
$\left\{0,\theta_{i,k} \right\}_{1\leq k\leq n_i}=\left\{\delta_{i+1}-\delta_i,0,\tau_{i,k}\right\}_{2\leq k\leq n_i}$ because
$\theta_{i,k}=\tau_{i,k}$ for $2\leq k \leq n_i$.
Thus
$\max\left\{0,\theta_{i,k} \right\}_{1\leq k\leq n_i}=\max\left\{\delta_{i+1}-\delta_i,0,\tau_{i,k}\right\}_{1\leq k\leq n_i}$.

\medskip
\noindent
If $h_{i,1}<\delta_{i}+1$, we have $\theta_{i,k}=\tau_{i,k}$ for $1\leq k \leq n_i$ and
$\delta_{i+1}-\delta_i\leq \tau_{i,1}$.
\\
So again
$\max\left\{0,\theta_{i,k} \right\}_{1\leq k\leq n_i}=\max\left\{\delta_{i+1}-\delta_i,0,\tau_{i,k}\right\}_{1\leq k\leq n_i}$.
\end{proof}
\begin{remark}\label{interprete_tau_i_k}
Let $\mathfrak{h}_i=\left(h_{i,1}^{l_{i,1}}, \dots,  h_{i,k}^{l_{i,k}}, \dots , h_{i,n_{i}}^{l_{i,n_{i}}}\right)$ be the $i$-th block of the hook code $\mathfrak{Q}(P)$.
Moving along the $i$-th diagonal of $P$ from top to bottom, we see that $\displaystyle{\sum_{j=k}^{j=n_i}l_{i,j}}$ is, by definition, the number of degree-$i$ hand monomials of $P$ below the $(l_{i,1}+\ldots+l_{i,k-1})$-th degree-$i$ hand monomial of $P$. Also, by definition, $h_{i,k}$ is the number of degree $i$ \textsl{horizontal-border monomials} of $P$ below the $(l_{i,1}+\ldots+l_{i,k-1}+1)$-th degree-$i$ hand monomial of $P$. \\
We may visualize the key integer $\tau_{i,k}$ related to $\beta_{i+1,0}(P)$ in the Ferrers diagram of $P$ by coloring the corresponding $\displaystyle{\sum_{j=k}^{j=n_i}l_{i,j}}$ hand monomials in red and the $h_{i,k}$ \textsl{horizontal-border monomials} in blue, in the next example (Figure \ref{tau_14_2_fig}).
\end{remark}
\begin{example}\label{tau_figure}
Let $T=(1,2,\ldots,12,13,12,6)$ and consider the partition $P$ of diagonal lengths $T$ given by $P=(14^2,12,11^2,10,7^2,5^3,4,3,1)$.
$T$ is a two-block Hilbert function.
The hook code of $P$ is $\mathfrak{Q}(P)=\left(\mathfrak{h}_{13},\mathfrak{h}_{14}\right)$. We have
$\delta_{13}=1$, $\delta_{14}=6$, $\delta_{15}=6$, $\mathfrak{h}_{13}(P)=(2^3,1^1,0^2)$ and $\mathfrak{h}_{14}(P)=(6^1,4^2,2^3)$.
\\
Note that $h_{13,1}=2=\delta_{13}+1$, $h_{14,1}=6<\delta_{14}+1$ and $\delta_{14}-\delta_{13}=5$.
\begin{itemize}
\item[$\bullet$] $\tau_{13,1}$, $\tau_{13,2}$ and $\tau_{13,3}$ are computed using the hook code block $\mathfrak{h}_{13}=(2^3,1^1,0^2)$.
\begin{itemize}
\item $\tau_{13,1}=3+1+2-2=4$. We may visualise $\tau_{13,1}$ by coloring the sequence $(2^3,1^1,0^2)$:
$(2^3,1^1,0^2)=(\textcolor{blue}{2}^{\textcolor{red}{3}},1^{\textcolor{red}{1}},0^{\textcolor{red}{2}})$;
$$
\begin{array}{c}
\tau_{13,1}=\text{\textsl{\textcolor{red}{sum of red integers} minus the \textcolor{blue}{blue integer}}}
\\
\displaystyle{
\tau_{i,k}=\textcolor{red}{\sum_{j=k}^{j=n_i}l_{i,j}}-\textcolor{blue}{h_{i,k}}
}
\end{array}
.
$$
\item $\tau_{13,2}=1+2-1=2$. $\tau_{13,2}$ can be visualised by coloring the subsequence $(1^1,0^2)$ of $(2^3,1^1,0^2)$:
$(2^3,1^1,0^2)=(2^3,\textcolor{blue}{1}^{\textcolor{red}{1}},0^{\textcolor{red}{2}})$, so $\tau_{13,2}=\text{\textsl{sum of red integers minus the blue integer}}$.
\item $\tau_{13,3}=2-0=2$, computed using $(2^3,1^1,0^2)=(2^3,1^1,\textcolor{blue}{0}^{\textcolor{red}{2}})$;
$\tau_{13,3}=\text{\textsl{sum of red integers minus the blue integer}}$.
\end{itemize}
We then find that $\beta_{14,0}(P)=\max\left\{\delta_{14}-\delta_{13},0,\tau_{13,1}, \tau_{13,2},\tau_{13,3}\right\}=5$.
\item[$\bullet$] to compute $\tau_{14,1}$, $\tau_{14,2}$ and $\tau_{14,3}$ we can use the same coloring method on the hook code block $\mathfrak{h}_{14}=(6^1,4^2,2^3)$.
\begin{itemize}
\item $\tau_{14,1}=1+2+3-6=0$; this is $\tau_{14,1}$=\textsl{sum of red integers minus the blue integer}, using the coloring
$(\textcolor{blue}{6}^{\textcolor{red}{1}},4^{\textcolor{red}{2}},2^{\textcolor{red}{3}})$.
\item $\tau_{14,2}=\textcolor{red}{2+3}-\textcolor{blue}{4}=1$ and $\tau_{14,3}=\textcolor{red}{3}-\textcolor{blue}{2}=1$ using the colorings
$(6^1,\textcolor{blue}{4}^{\textcolor{red}{2}},2^{\textcolor{red}{3}})$, $(6^1,4^2,\textcolor{blue}{2}^{\textcolor{red}{3}})$.
\end{itemize}
We then find that $\beta_{15,0}(P)=\max\left\{\delta_{15}-\delta_{14},0,\tau_{14,1}, \tau_{14,2},\tau_{14,3}\right\}=1$.
\end{itemize}
We illustrate $\tau_{14,2}$ in Figure \ref{tau_14_2_fig} by coloring the degree-$i$ hand monomials and the degree-$i$ \textsl{horizontal-border monomials} as suggested in Remark \ref{interprete_tau_i_k}
\begin{figure}[!h]
\begin{center}
	\begin{tikzpicture}[scale=.65]
	\draw[dashed,very thin] 	(0,0)--(14,0)
						(0,-1)--(14,-1)
						(0,-2)--(12,-2)
						(0,-3)--(11,-3)
						(0,-4)--(11,-4)
						(0,-5)--(10,-5)
						(0,-6)--(7,-6)
						(0,-7)--(7,-7)
						(0,-8)--(5,-8)
						(0,-9)--(5,-9)
						(0,-10)--(5,-10)
						(0,-11)--(4,-11)
						(0,-12)--(3,-12)
						(0,-13)--(1,-13)
						(0,0)--(0,-14)
						(1,0)--(1,-13)
						(2,0)--(2,-13)
						(3,0)--(3,-12)
						(4,0)--(4,-11)
						(5,0)--(5,-8)
						(6,0)--(6,-8)
						(7,0)--(7,-6)
						(8,0)--(8,-6)
						(9,0)--(9,-6)
						(10,0)--(10,-5)
						(11,0)--(11,-3)
						(12,0)--(12,-2)
						(13,0)--(13,-2);
	\draw[thin,red,fill=red!8] (10,-4) rectangle (11,-5)
							(9,-5) rectangle (10,-6)
							(4,-10) rectangle (5,-11)
							(3,-11) rectangle (4,-12)
							(2,-12) rectangle (3,-13);
	\draw[thin, blue,fill=black!5,rounded corners=2mm] (8,-6) rectangle (9,-7)
					     	(6,-8) rectangle (7,-9)
						(1,-13) rectangle (2,-14)
					  	(0,-14) rectangle (1,-15);
	\draw[red] node[font=\tiny] at (10.5,-4.5) {$x^{10}y^4$}
			  node[font=\tiny] at (9.5,-5.5) {$x^9y^5$}
			  node[font=\tiny] at (4.5,-10.5) {$x^4y^{10}$}
			  node[font=\tiny] at (3.5,-11.5) {$x^3y^{11}$}
			  node[font=\tiny] at (2.5,-12.5) {$x^2y^{12}$};
	\draw[blue] node[font=\tiny] at (8.5,-6.5) {$x^8y^6$}
			    node[font=\tiny] at (6.5,-8.5) {$x^6y^8$}
			    node[font=\tiny] at (1.5,-13.5) {$xy^{13}$}
			    node[font=\tiny] at (0.5,-14.5) {$y^{14}$};
\draw[thick] (14,0)--(14,-2)--(12,-2)--(12,-3)--(11,-3)--(11,-5)--(10,-5)--(10,-6)--(7,-6)--(7,-8)--(5,-8)--(5,-11)--(4,-11)--(4,-12)--(3,-12)--(3,-13)--(1,-13)--(1,-14)--(0,-14);
\end{tikzpicture}
\caption{$P=(14^2,12,11^2,10,7^2,5^3,4,3,1)$ : $\tau_{14,2}=\textcolor{red}{2+3}-\textcolor{blue}{4}=1$ }
\label{tau_14_2_fig}
\end{center}
\end{figure}
\end{example}
\pagebreak
\begin{theorem}\label{extra_gens_in_given_degree}
Let $P\in \mathcal P(T), T=\left(1,\dots ,d,t_d,\dots ,t_{\sf j},0\right)$ have hook code
 $\mathfrak{Q}(P)=\left(\mathfrak{h}_d,\dots ,\mathfrak{h}_{i}, \dots, \mathfrak{h}_{{\sf j}}\right)$.
 Suppose $\mathfrak{h}_i=\left(h_{i,1}^{l_{i,1}}, \dots,  h_{i,k}^{l_{i,k}}, \dots , h_{i,n_{i}}^{l_{i,n_{i}}}\right)$ is the $i$-th block
 of the hook code $\mathfrak{Q}(P)$ of $P$.
 Let  $I$ be a generic element of the cell ${\mathbb V}(E_P)$ and ${\mathcal G}(I)$ a minimal set of generators of $I$, and $\beta_{i,0}(P)$ the number of degree-$i$ generators. We denote by $\overset{\to}{\kappa}(P)=\beta_0(P)=\left(\beta_{d,0}(P),\ldots ,\beta_{{\sf j}+1,0}(P)\right)$.
 For $d\leq m \leq {\sf j}+1$, let $$N_m=\#\left\{ f\in {\mathcal G}(I), \text{ such that degree}(f) \leq m \right\}.$$
 Then $N_d=d+1-t_d$, and using the previously defined numbers $r_{i,k}$ and $g_{i,k}$, for $i\in\left[d,{\sf j}\right]$, we have for $\beta_{i+1,0}(P)=N_{i+1}-N_i$,
 \begin{equation}\label{beta0eq}
 \begin{array}{ll}
 N_{i+1}-N_{i}&=\max\left\{0,g_{i,k}-r_{i,k} \right\}_{1\leq k \leq n_i} \\
 &=\max\left\{\delta_{i+1}-\delta_i,0,\tau_{i,k}\right\}_{1\leq k\leq n_i}.
 \end{array}
 \end{equation}

\end{theorem}
\begin{proof}
\noindent
By standard basis construction techniques (see Theorem I.1.9 of \cite{brian}, Proposition 2 and Proposition 3 of \cite{brian-gal}), one can  first see that if $r_{i,n_i}\geq g_{i,n_i}$, then there are enough degree
$i+1$ relations to kick out all of the $g_{i,n_i}$ generators that are just above these relations. Also, if $r_{i,n_i}<g_{i,n_i}$,
then we need $g_{i,n_i}-r_{i,n_i}$ extra generators whose leading terms are corner monomials of $E_P$ below the
$\left(l_{i,1}+ \dots + l_{i,n_i-1}+1\right)$-th degree $i$ hand of $P$.

\medskip
\noindent
It is clear that if for all $k$ ($1\leq k\leq n_i$) we have $r_{i,k}\geq g_{i,k}$, then we can kick out all degree $i+1$ generators whose leading terms are degree $i+1$ corner-monomials of $E_P$. \\
Now, suppose there exists an integer $k$ such that $r_{i,k}<g_{i,k}$. Then we can inductively consider the following sets and numbers.
$$
\begin{array}{lr}
S_0=\left\{k\in {\mathbb N}, \quad  r_{i,k}<g_{i,k}\right\}, & s_0=\max(S_0);
\\
S_1=\left\{k\in {\mathbb N},  \quad k<s_0, \quad r_{i,k}-r_{i,s_0}<g_{i,k}-g_{i,s_0}\right\}, & s_1=\max(S_1);
\\
\dotfill &\dotfill ;
\\
S_{q}=\left\{k\in {\mathbb N},  \quad k<s_{q-1}, \quad r_{i,k}-r_{i,s_{q-1}}<g_{i,k}-g_{i,s_{q-1}}\right\}, & s_q=\max(S_q);
\\
S_{q+1}=\emptyset. &
\end{array}
$$
The meaning of the sets $S_0, \cdots, S_q$ and the numbers $s_0, \cdots, s_q$ is the following:
\begin{itemize}
\item[$\bullet$] First, we have $r_{i,s_{0}+1}\geq g_{i,s_{0}+1}, \cdots, r_{i,n_i}\geq g_{i,n_i}$ and $r_{i,s_0}<g_{i,s_0}$. So we need
$g_{i,s_0}-r_{i,s_0}$ generators in the $(n_i-s_0+1)$-th part of the chain
$G_{i,n}\subset G_{i,n-1} \subset \cdots \subset G_{i,k+1}\subset G_{i,k} \subset \cdots \subset G_{i,1}$. This means we have used all the relations
in the $(n_i-s_0+1)$-th part of the chain
$R_{i,n}\subset R_{i,n-1} \subset \cdots \subset R_{i,k+1}\subset R_{i,k} \subset \cdots \subset R_{i,1}$.
\item[$\bullet$] Since we have used all the relations in $R_{i,n_i-s_0+1}$, if we are looking for more extra generators,
the next step is to consider the chains of inclusions
$$
 \begin{array}{l}
 (R_{i,n_i-s_0}-R_{i,n_i-s_0+1}) \subset \cdots \subset (R_{i,1}- R_{i,n_i-s_0+1})\\
 (G_{i,n_i-s_0}-G_{i,n_i-s_0+1}) \subset \cdots \subset (G_{i,1}-G_{i,n_i-s_0+1})
 .
 \end{array}
 $$
If the set $S_1=\left\{k\in {\mathbb N},  \quad k<s_0, \quad r_{i,k}-r_{i,s_0}<g_{i,k}-g_{i,s_0}\right\}$ is not empty, then we
set $s_1=\max(S_1)$ and continue looking for extra generators until $S_{q+1}=\emptyset$ and $S_{q}\neq \emptyset$ for some index $q$.
\end{itemize}
By construction, the number of degree $i+1$ extra generators needed is
$$
 g_{i,s_0}-r_{i,s_0}+\sum_{j=1}^{j=q}\left((g_{i,s_j}-g_{i,s_{j-1}})-(r_{i,s_j}-r_{i,s_{j-1}})\right)=g_{i,s_q}-r_{i,s_q}
 .
$$
It is then clear that
$$
N_{i+1}-N_{i}=g_{i,s_q}-r_{i,s_q}=\max\left\{0,g_{i,k}-r_{i,k} \right\}_{1\leq k \leq n_i}
.
$$
\end{proof}

\begin{remark}\label{lower_bound_for_degree_i+1_gens}
Using the notation of Lemma \ref{kappaTlem} and Theorem \ref{extra_gens_in_given_degree} one has
\begin{enumerate}[(i).]
\item $\left[\delta_{i+1}-\delta_i\right]^{+}\leq \max\left\{\delta_{i+1}-\delta_i,0,\tau_{i,k}\right\}_{1\leq k\leq n_i}=N_{i+1}-N_{i}$, so the number of degree $i+1$ generators of $I\in {\mathbb V}(E_P)$ is at least $\left[\delta_{i+1}-\delta_i\right]^{+}$ (this is statement i. of Lemma \ref{kappaTlem}).
\item Suppose $P$ is the partition associated to the generic cell of $\G_T$. Then for the $i$-th block
$\mathfrak{h}_i=\left(h_{i,1}^{l_{i,1}}, \dots,  h_{i,k}^{l_{i,k}}, \dots , h_{i,n_{i}}^{l_{i,n_{i}}}\right)$ of the hook code $\mathfrak{Q}(P)$ of $P$, we have $n_i=1$, $h_{i,1}=\delta_i+1$ and $l_{i,1}=\delta_{i+1}$, that is $\mathfrak{h}_i=\left(\delta_i+1\right)^{\delta_{i+1}}$. In this case, $g_{i,1}=\delta_{i+1}$, $r_{i,1}=\delta_i$ and formula \ref{beta0eq} of Theorem \ref{extra_gens_in_given_degree}
gives $N_{i+1}-N_{i}=\left[\delta_{i+1}-\delta_i\right]^{+}$ (this is statement ii. of Lemma \ref{kappaTlem}).
\end{enumerate}
\end{remark}

\begin{corollary}
Let $T_i=(1, \cdots, d-1,d,t_d,0)$ be a single-block Hilbert function where $d=\delta_i+\delta_{i+1}$ and $t_d=\delta_{i+1}$.
Suppose $P_i$ is a partition of diagonal lengths $T_i$ and difference-one hook code  $\mathfrak{Q}(P_i)=\mathfrak{h}_i$ with
$$
\begin{array}{l}
\mathfrak{h}_i=\left(h_{i,1}^{l_{i,1}}, \dots,  h_{i,k}^{l_{i,k}}, \dots , h_{i,n_{i}}^{l_{i,n_{i}}}\right); \\
\delta_i+1\geq h_{i,1}> \dots > h_{i,n_{i}}\geq 0 \quad \text{and} \quad \displaystyle{\sum_{k=1}^{k=n_i}l_{i,k}=\delta_{i+1}}.
\end{array}
$$
Then $\kappa(P_i)=\delta_i+1+\max\left\{0,g_{i,k}-r_{i,k} \right\}_{1\leq k \leq n_i}$.
\end{corollary}

\begin{corollary}\label{multicountcor}
Let $T=(1, 2, \cdots , d-1, t_d, \cdots , t_{\sf j}, 0)$ be a Hilbert function and $P$ a partition of diagonal lengths $T$. Let  $I$ be a generic element of the cell ${\mathbb V}(E_P)$ and ${\mathcal G}(I)$ a minimal set of generators of $I$. For $d\leq i\leq {\sf j}+1$, let
$N_i=\#\left\{ f\in {\mathcal G}(I), \text{degree}(f) \leq i \right\}$. Then
\begin{enumerate}[(a)]
\item for any $i\in [d,{\sf j}]$, $N_{i+1}-N_{i}=\kappa(P_i)-(\delta_i +1)$.
\item $\kappa(P)=\displaystyle{\#{\mathcal G}(I)=N_{\sf j}+1}=N_d+\sum_{i=d}^{i={\sf j}}\left(\kappa(P_i)-(\delta_i +1)\right)$.
\end{enumerate}
\end{corollary}

\begin{remark}\label{empty-component-contribution}
Note that if $t_i=t_{i+1}$, then it is clear that $N_{i+1}-N_{i}=0$. Also, we found  (Proposition \ref{kappa-of-empty-bloc}) that when $t_i=t_{i+1}$, we have
$\kappa(P_i)=\delta_i +1$. So the empty hook blocs contribute to zero in the sum computing $\kappa(P)$ in Corollary \ref{multicountcor}.
\end{remark}
\medskip

\begin{proposition}\label{E_P_Betti_sequence}
Let $P$ be a partition of diagonal lengths $T=(1, 2, \cdots , d-1, t_d, \cdots , t_{\sf j}, 0)$ and suppose
$\mathfrak{h}_i=\left(h_{i,1}^{l_{i,1}}, \dots,  h_{i,k}^{l_{i,k}}, \dots , h_{i,n_{i}}^{l_{i,n_{i}}}\right)$ is the $i$-th block
 of the hook code $\mathfrak{Q}(P)$ of $P$.  \\
 Denote by $b_{i+1}(E_P)$ the number of degree $i+1$ corner-monomials (generators) of $E_P$. Then we have
 $$
 \begin{array}{l}
 b_d=d+1-t_d,\\
 b_{i+1}(E_P)=\delta_{i+1}-n_i+\max\{1-h_{i,n_i},0\}+\max\{h_{i,1}-\delta_i,0\}.

 \end{array}
 $$
\end{proposition}
\begin{proof}
The number of degree $d$ corner-monomials of $E_P$ is of course $d+1-t_d$. By Definition \ref{degree_i+1_rel_gen_sequences} we have $b_{i+1}(E_P)=g_{i,1}$. The proof of the Proposition then follows directly from Lemma \ref{degree_i+1_rel_gen_formulas}  and Remark \ref{compact_formulas_for_gik_rik}.
\end{proof}

We note that Theorem~\ref{componentTheorem} is in fact a restatement of Corollary~\ref{multicountcor} above.
\par
Recall from Definition \ref{special-partition-definition} that a partition $P$ of diagonal lengths $T$ is special if $\kappa(P)\neq \kappa(T)$. In other words, $P$ is special if $\kappa(P)$ does not have the minimum value $\kappa(T)$ possible for partitions of diagonal lengths $T$, from Equation \ref{kappaTeq}. The following immediate corollary of  Theorem~\ref{componentTheorem} gives a necessary and sufficient condition for a partition $P$ to be special. Recall that Corollary \ref{special1cor} specifies when  a single-block partition is special.

\begin{theorem}
[Component Theorem for $P$ special]\label{componentthm}
Assume that $T$ is a Hilbert function of height $d$ and socle degree ${\sf j}$,
 and that the partition $P$ of diagonal lengths $T$ decomposes into single block partitions $P_d, \dots
, P_{\sf j}$. Then $P$ is special if and only if  $P_i$ is special, for some $i\in [d, {\sf j}]$.
\end{theorem}

\begin{proof}
By Theorem~\ref{componentTheorem}, the value of $\kappa(P)$ is minimum - equal to $\kappa(T)$ - if and only if $\kappa(P_i)$ is minimum for all $i\in [d,{\sf j}]$. Thus $P$ is non-special if and only if  at least one component $P_i$ is non-special for an integer $i\in [d,{\sf j}]$.
\end{proof}
\begin{figure}[t!]
\centering
\scalebox{0.6}{\begin{ytableau}
\textcolor{gray}{\bullet} & &&& \textcolor{gray}{\bullet} &\textcolor{gray}{\bullet} &&& \textcolor{gray}{\bullet}&&&&&*(black! 20)\textcolor{gray}{\bullet} &*(black! 50)\cr
$ $ & &&&&&&&&&&\cr
$ $ &&&\circ &&&\circ &\circ &&&&*(black! 20)\cr
\textcolor{gray}{\bullet} &&&&\textcolor{gray}{\bullet} & \textcolor{gray}{\bullet}&&&\textcolor{gray}{\bullet} && *(black! 20)&*(black! 50)\cr
$ $ & \bullet &   &   &   &   &   &   &   &*(black! 20)\bullet &*(black! 50) &*(black! 100)\cr
$ $  &\bullet&&& & &&&*(black! 20) &*(black! 50)\bullet&*(black! 100)\cr
$ $& &&&&&\cr
$ $& &&&&\cr
$ $& &&  \circ &&*(black! 20)\cr
$ $& && \circ  &*(black! 20)\cr
$ $& &\cr
$ $& &*(black! 20)\cr
\textcolor{gray}{\bullet}& *(black! 20)&*(black! 50)\cr
*(black! 20)&*(black! 50)\bullet&*(black! 100)\cr
\end{ytableau}}\caption{Ferrers diagram of the partition $P=(15, 12^4, 11, 7, 6^2, 5, 3^4)$ of diagonal lengths $T=\left(1,2,\dots ,13,10_{13},6_{14},3_{15},0\right)$ and difference-one hook code $\mathfrak{Q}(P)=\left((3,1^2, 0)_{13},(5,4,1)_{14}, (2^2,1)_{15}\right)$. Each labeled box represents a hook corner, it is labeled by a $\circ$ if its hand degree is 13, with a \textcolor{gray}{$\bullet$} if the hand degree is 14, and with a dark $\bullet$ when the hand degree is 15 (Example \ref{gensEx}).}\label{3blockdiag}
\end{figure}

\begin{example}\label{gensEx}
Consider the partition $P=(15, 12^4, 11, 7, 6^2, 5, 3^4)$ from Example \ref{1stgensEx}. Then $P$ has diagonal lengths $T=\left(1,2,\dots ,13,10_{13},6_{14},3_{15},0\right)$ and hook code (see Figure \ref{3blockdiag}) $$\mathfrak{Q}(P)=\left((3,1^2, 0)_{13},(5,4,1)_{14}, (2^2,1)_{15}\right).$$ Then, as we saw in Example \ref{1stgensEx}, $P$ can be decomposed into the following three single-block partitions. Partition $P_{13}=(7^2, 5, 4^2, 3, 1^2)$ of diagonal lengths $T_{13}=(1, \dots, 7, 4,0)$ and hook code $\mathfrak{Q}(P_{13})=(3, 1^2, 0)$,
partition $P_{14}=(8, 6^2,4, 3, 2^2)$ of diagonal lengths $T_{14}=(1, \dots, 7, 3,0)$ and hook code $\mathfrak{Q}(P_{14})=(5, 4, 1)$,
and partition $P_{15}=(6, 5^2, 4, 2^2)$ of diagonal lengths $T_{15}=(1, \dots, 6, 3,0)$ and hook code $\mathfrak{Q}(P_{15})=(2^2,1)$.

By Theorem~\ref{kappathm}, we have
\[
\kappa(P_{13})=6,\quad \kappa(P_{14})=5, \mbox{ and }
\kappa(P_{15})=5.
\]

Thus by Theorem~\ref{componentTheorem}, we have $$\kappa(P)=6+5+5-(10-3)-2=7.$$

Let $I$ is a generic ideal in the cell $\mathbb{V}(E_P)$. Then a minimal set of generators for $I$ consists of seven generators. Of these seven generators, four have degree 13, two have degree 14, and one has degree 15.
\end{example}
\subsubsection{Elementary and non-elementary Hilbert functions.}

We say that a sequence $T$ satisfying Equation \eqref{Teq} is \emph{elementary} if there is no integer $i\in [d,{\sf j}]$ such that
$t_i=t_{i+1}<d$ \cite[\S 4Ai]{IY}; then we also say that $\G_T$ is elementary. It is well known that when a Hilbert function $T=H(R/I)$ satisfies $t_i=t_{i+1}=s<d$ then there is a form $f\in R_s$ such that
\begin{equation}\label{Tconsteq} I_i=fR_{i-s} \text { and }I_{i+1}=f\cdot R_{i+1-s}.
\end{equation}  If follows that
$f| I_u$ for $u\le i+1$. This is usually shown using the properties of $\tau(V)=\dim_{\sf k} R_1V-\dim_{\sf k}V $ for vector subspaces $V\subset R_i$: this integer is the number of generators of an ``ancestor ideal'' $I=(V)\oplus_{u=1}^i V:R_u$, and $\tau (I_i)=1$ when $t_i=t_{i+1}$: see \cite[p. 56]{Ia} or \cite[Lemma 2.2]{Ia2}.\par
We will define implicitly in the next Theorem ``elementary factors'' $T(i)$ of Hilbert sequences $T$ which have constant subsequences of height $s<d$. These factors have no relation with the single block components $T_i$ for each $T$, defined in Equation \eqref{HFdecomposition}, and a major topic for us. In fact
if $T$ splits into elementary components $T(i)$ they are not usually single-block.
\begin{lemma}\label{elemlem}\cite[Lemma 4.2]{IY} There is a decomposition of $\G_T$ as a product
$$\G_T=\prod_k G_{T(k)} \text { for $T(k)$ elementary}.$$
\end{lemma}
\begin{proof} Assume there is a single maximal consecutive subsequence
$t_i=t_{i+1}=\cdots =t_{i+k}=s$ with $k\ge 1$ and $s<d$.  Then consider $T(1)=(1,2,\ldots,s_{s-1},s,\ldots  ,s_{i+k},t_{i+k+1},\ldots ,t_{\sf j})$, and $T(2)$ defined by $T(2)_{u}=t_{u+s}-s$ for $u\le i-s$.  Let $p_I\in \G_T$ be a point parametrizing the graded ideal $I$ such that $A=R/I$ satisfies $H(A)=T$. Then we let $I(1)=(f_s,I)$. We have $I_{u+s}=f_sV_{u}$ for $0\le u\le i-s$: we define $I(2)_u=V_u$ for $u\in [0, i-s]$. Then the pair $(I(1),I(2))$ determines $I$ and conversely. This proves the Lemma for $k=2$, it is straightfoward to extend it to $k\ge 2$.
\end{proof}

\begin{remark}\label{kappa_cell1xcell_2_rem}\par
Let $T=(1, 2, \cdots , d, t_d, \cdots , t_{\sf j}, 0)$ be a Hilbert function as in Equation \ref{Teq} and $P$ a partition of diagonal lengths $T$.
Suppose there is an integer $i\in\left[d,{\sf j}-1 \right]$ such that $t_i=t_{i+1}=\cdots =t_{i+k}$ with $k\geq 1$ and $s<d$.
Let  $I$ be a generic element of the cell ${\mathbb V}(E_P)$ and ${\mathcal G}(I)$ a minimal set of generators of $I$. Let
${\mathcal G}(I)_1=\left\{ f\in {\mathcal G}(I), \text{degree}(f) \geq i+1 \right\}$ and ${\mathcal G}(I)_2=\left\{ f\in {\mathcal G}(I), \text{degree}(f) \leq i \right\}$.  Setting $m_1=\left|{\mathcal G}(I)_1\right|$ and $m_2=\left|{\mathcal G}(I)_2\right|$, we get  $\kappa(P)=m_1+m_2$. We know from Equation \eqref{Tconsteq} that there is a degree $s$ form $f_s$ such that $f_s$ divides each of the elements $f_1, \ldots , f_{m_2}$ of ${\mathcal G}(I)_2$.
Let $I(1)=(f_s,I)$ and $I(2)=(I:f_s)$. Then $I(1)$ is a generic element of a cell ${\mathbb V}(E_{P(1)})$ and $I(2)$ is a generic element of a
cell ${\mathbb V}(E_{P(2)})$. It is clear that $\kappa({P(1)})=m_1+1$ and $\kappa({P(2)})=m_2$, so $\kappa(P)=\kappa({P(1)})+\kappa({P(2)})-1$.
\end{remark}

\begin{proposition}\label{kappa_prod_elem_cells_prop}
Suppose that the variety $\G_T$ decomposes as $\displaystyle{\G_T=\prod_{k=1}^{k=r} G_{T(k)}}$ with each $T(k)$ elementary.
Then any cell ${\mathbb V}(E_P)$ of $\G_T$ decomposes as
$$\displaystyle{{\mathbb V}(E_P)=\prod_{k=1}^{k=r} {\mathbb V}(E_{P(k)})} \text { for $P(k)$ a partition of diagonal lengths $T(k)$}.$$
Also $\displaystyle{\kappa(P)=\sum_{k=1}^{k=r}\kappa(P(k))-r+1} $.
\end{proposition}

\begin{proof}
The Proposition follows from Remark \ref{kappa_cell1xcell_2_rem}.
\end{proof}

\begin{example}\label{flatHFexample}(See Figure \ref{5.8figure}.)
Let $P=\left(10^2,4,3^2,2^5\right)$ be the partition of diagonal lengths $$T=\left(1,2,\dots ,6_5,5_6,4_7,4_8,4_9,2_{10},0\right),$$ and difference-one hook code $$\mathfrak{Q}(P)=\left((0)_6,(1,0)_9,(2,1)_{10}\right). $$
Let $I$ be a generic element of $\mathbb{V}(E_P)$.
The elementary components of $T$, explained in Lemma \ref{elemlem} and Remark \ref{kappa_cell1xcell_2_rem} are
\begin{equation*}
T(1) = (1,2,3,4,4_4,4_5,\dots,4_9,2_{10},0),\quad \text{and}\hspace*{2mm} T(2)=(1,2,1).
\end{equation*}
As it is explained in Remark \ref{kappa_cell1xcell_2_rem} we let $I(1) = (f_4,I)$ be a generic element in the cell $\mathbb{V}(E_{P(1)})$ with the Hilbert function $T(1)$. Also $I(2)=(I\colon f_4)$ is a generic element in the cell $\mathbb{V}(E_{P(2)})$ with the Hilbert function $T(2)$. We have $P(1) = (10^2,2^8)$ and $P(2)=(2,1^2)$ which are subpartions of $P$ in different colors in Figure \ref{5.8figure}.
We easily see that $\kappa(P(1)) = \kappa(P(2))=3$ and therefore by Proposition \ref{kappa_prod_elem_cells_prop} we get $$\kappa(P) = \kappa(P(1)) + \kappa(P(2))-1 =5.$$

We could also compute $\kappa(P)$ by decomposition of $P$ and  $T$ into single-block components, see Equation \ref{HFdecomposition}.
Single-block component partitions $P_6,\dots ,P_{10}$ of diagonal lengths $T_{6},\dots ,T_{10}$ as follows,
$$
P_6=\left(2,1,1\right), \hspace*{0.2cm}P_7=(1), \hspace*{0.2cm}P_8=\left(0\right), \hspace*{0.2cm} P_9=\left(3,1,1\right)\hspace*{0.2cm}\text{and}\hspace*{0.2cm} P_{10}=\left(4,4,2,2\right),$$
$$
T_6=\left(1,2,1\right), \hspace*{0.2cm}T_7=(1,0), \hspace*{0.2cm}T_8=\left(0\right), \hspace*{0.2cm} T_9=\left(1,2,2,0\right)\hspace*{0.2cm}\text{and}\hspace*{0.2cm} T_{10}=\left(1,2,3,4,2,0\right).$$
The hook codes of $P_6$, $P_9$ and $P_{10}$ are  $\mathfrak{h}_6=(0)$, $\mathfrak{h}_9=(1,0)$ and $\mathfrak{h}_{10}=(2,1)$ respectively.
Using Theorem \ref{kappathm} we get that
$$\kappa(P_6)=\kappa(P_9)=\kappa(P_{10})=3,
$$
and for $P_7=\Delta_1$ and $P_8=\Delta_\emptyset$ by Remark \ref{empty_hook_code_block_partition}, we conclude that
$$
\hspace*{0.2cm}\kappa(P_7)=2,\hspace*{0.2cm} \kappa(P_8)=1.
$$
Therefore, Theorem \ref{componentthm} implies that

$$\kappa(P)=3+2+1+3+3-(4-2)-(10-6)=5$$

We also note that of these five generators, two generators have degree 6 and one generator has degree $7$ (corresponding to generators of $P_6$), and two have degree 10.\par
Note that $\dim \G_{T_{10}}=2(3)=6, \dim \G_{T_9}=(2)(1)=2$, and $\dim \G_T=8$, since $\G_T$ is fibred over $\mathbb P_4$ parametrizing the generator $f_4$ of $I_6$ by a Grassmannian $\Grass(2,4)$ parametrizing $I_{10}/f_4R_6$, a two-dimensional subspace of $R_{10}/f_4R_6$, which has dimension four.
\par
\end{example}
\begin{figure}
\centering
\scalebox{0.6}{\begin{ytableau}
*(blue! 15) &*(blue! 15)&*(blue! 15)&*(blue! 15)&*(blue! 15)&*(blue! 15)&*(blue! 15)&*(blue! 15)&*(blue! 15)&*(blue! 15)\cr
*(blue! 15) &*(blue! 15)&*(blue! 15)&*(blue! 15)&*(blue! 15)&*(blue! 15)&*(blue! 15)&*(blue! 15)&*(blue! 15)&*(blue! 15)\cr
*(blue! 15)&*(blue! 15)&*(red! 20)&*(red! 20)\cr
*(blue! 15)&*(blue! 15)&*(red! 20)\cr
*(blue! 15)&*(blue! 15)&*(red! 20)\cr
*(blue! 15)&*(blue! 15)\cr
*(blue! 15)&*(blue! 15)\cr
*(blue! 15)&*(blue! 15)\cr
*(blue! 15)&*(blue! 15)\cr
*(blue! 15)&*(blue! 15)\cr
\end{ytableau}}\caption{Diagram of the partition in Example \ref{flatHFexample}. The
elementary partitions $P(1)$ and $P(2)$ are colored blue and red respectively.}\label{5.8figure}
\end{figure}

\newpage
\section{Number of cells of special multiblock partitions.}\label{countMultiSection}
 Using Corollary \ref{singlecountcor} and Theorem \ref{componentTheorem} we are able to count the number of multiblock partitions with a given number of generators.

\begin{theorem}\label{multicountthm}
Assume that $T=(1, \dots, d, t_d, \dots, t_{\sf j},0)$ and for $d\leq i \leq \sf j$, let $T_i=(1, \dots,t_{i-1}-t_{i+1}, t_{i}-t_{i+1},0)$. Then for every positive integer $k$, the number of partitions $P$ of diagonal lengths $T$ and $\kappa(P)=k$, denoted by $\mu(T,k)$, satisfies
\begin{equation}\label{multicountEq}
\mu(T,k)=\sum_{(k_d,\dots,k_{\sf j})\in Q_k} \left(\prod_{i=d}^{\sf j}\mu(T_i,k_i)\right),
\end{equation}
where $Q_k=\{(k_d, \dots, k_{\sf j})\in \mathbb{Z}^{\mathsf{j}+1-d}\, |\, k_d+\dots+k_{\sf j}=k+(t_d-d)-(t_{\sf j}-{\sf j})\}.$
\end{theorem}

\begin{proof}
This is an immediate consequence of Theorem~\ref{componentTheorem}. Also recall that Corollary~\ref{singlecountcor} provides an explicit formula for $\mu(T_i,k_i)$, for every $d\leq i \leq \sf j$.

\end{proof}
\begin{remark}
For each $i\in [d,\sf j]$, by Corollary~\ref{singlecountcor}, $\mu(T_i,k_i)$ is non-zero if and only if $\max\{t_i-t_{i+1}+1, t_{i-1}-t_i+1\}+1\leq k_i \leq t_{i-1}-t_{i+1}+1$. Thus in Equation~(\ref{multicountEq}) we are effectively taking the sum over the points in the hyperplane defined by $k_d+\dots+k_{\sf j}=k+(t_d-d)-(t_{\sf j}-\sf j)$ in the hyper cubes obtained by the product of line segments of the form  $[\max\{t_i-t_{i+1}, t_{i-1}-t_i\}+1, t_{i-1}-t_{i+1}+1]$ in $\mathbb{Z}^{\mathsf{j}+1-d}$.
\end{remark}
Recall that $\mathcal{P}(T)$ is the set of all partitions of diagonal lengths $T$. Denote by $A$ the cardinality of $\mathcal{P}(T)$. We have from \cite[Theorem 3.30]{IY}, or as a consequence of Equation \ref{count2eq} below, that
\begin{equation}\label{cardinalityPTeq}
A=\prod_{d\le i\le {\sf j}}\binom{t_{i-1}-t_{i+1}+1}{t_i-t_{i+1}}.
\end{equation}
A refinement, grading by the dimension of the cells, gives the Betti numbers of $G_T$.
Recall from Definition~\ref{special-partition-definition} that a partition $P$ of diagonal lengths $T$ is called special if $\kappa(P)>\kappa(T)$, and denote by $S$ the number of
special partitions of diagonal lengths $T$.\par
\noindent Using Definition \ref{Pidef1}, we decompose a partition $P$ of diagonal lengths $T=(1,2,\dots ,d, t_d,\dots , t_{\sf j},0)$ into ${\sf j}+1-d$ single block partitions, $P_d,\dots ,P_{\sf j}$, where for each $d\leq i\leq \sf j $, the diagonal lengths of $P_i$ is the sequence \linebreak$T_i=\left(1,\dots ,t_{i-1}-t_{i+1},t_i-t_{i+1},0\right)$.
For each $d\leq i\leq \sf j$, we denote the total number of partitions of diagonal lengths $T_i$ by $A_i$ (see Lemma \ref{count1blockcor} and Equation \eqref{count2eq} below) and the number of special partitions of diagonal lengths $T_i$ by $S_i$.
 The number of special partitions is equal to $\sum_{k>\kappa(T)}{\mu(T,k)}$, where $\mu(T,k)$ is described in the above theorem.\par
\noindent In the following, we provide the number of special partitions of diagonal lengths $T=\left(1,2,\dots ,d,t_d,\dots ,t_{\sf j},0\right)$, using the inclusion-exclusion principal.
\begin{corollary}[Number of special partitions]\label{countspecialcor}
The number of special partitions of diagonal lengths $T=\left(1,2,\dots ,d,t_d,\dots ,t_{\sf j},0\right)$ is equal to
\begin{equation}\label{specialmultiblock}
S=\sum_{i=1}^{\mathsf{j}-d+1}(-1)^{i+1}\left(\sum_{\lambda\subseteq \{d,\dots ,\mathsf{j}\}, \vert\lambda\vert=i}S_\lambda A_{\{d,\dots ,{\sf {j}}\}\setminus \lambda}\right),
\end{equation}
where $S_\lambda=\prod_{i\in \lambda} S_i$ and $A_{\{d,\dots ,{\sf j}\}\setminus \lambda}=\prod_{i\in {\{d,\dots ,{\sf{j}}\}\setminus \lambda}}A_i$.
\end{corollary}
\begin{proof}
Theorem \ref{componentthm} implies that $P$ is special if and only if $P_i$ is special for some $i\in[d,\sf j]$. \\
Note that for each $i\in [d, \sf j]$ the number of partitions of diagonal lengths $T_i$ is equal to
\begin{equation}\label{count2eq} A_i=\binom{t_{i-1}-t_{i+1}+1}{t_i-t_{i+1}}.
\end{equation}
On the other hand,
 Theorem \ref{countingthm} provides the number of special single block partitions. Using Equation~\ref{numberspecial1eq} for each $d\leq i\leq \sf j$ we obtain the number of special partitions of diagonal lengths $T_i$ as the following
\begin{equation}\label{count3eq}
S_i=\binom{t_{i-1}-t_{i+1}+1}{t_i-t_{i+1}-\delta_i-1},
\end{equation}
where $\delta_i=\max\{2t_i-2t_{i+1}-t_{i-1}+t_{i+1},0\}=\max\{2t_i-t_{i+1}-t_{i-1},0\}$.\\
\noindent Now using the inclusion-exclusion principal we get the equality of Equation~\eqref{specialmultiblock}.
\end{proof}

As a consequence of the above Theorem, we recover a result of \cite[Theorem 3.7]{AIK} providing the number of complete intersection Jordan types $P\in \mathcal P(T)$. Recall that a complete intersection Jordan type of diagonal lengths $T$ is a partition $P$ of diagonal lengths $T$ such that $\kappa(P)=2$.
\begin{corollary}\label{CIJordantypecor}
\begin{itemize}
\item[$(a)$]The number of complete intersection Jordan types of diagonal lengths
\begin{equation*}
T=\left(1_0,2_1,\dots ,(d-1)_{d-2},d_{d-1},(d-1)_{d},\dots , 2_{2d-3},1_{2d-2}\right)
\end{equation*}
is equal to $2^{d-1}$.
\item[$(b)$]The number of complete intersection Jordan types with diagonal lengths
\small\footnotesize
\begin{equation*}
T=\left(1_0,2_1,\dots ,(d-1)_{d-2},d_{d-1}, \dots , d_{d+k-2},(d-1)_{d+k-1},\dots , 2_{2d-4+k},1_{2d-3+k}\right)
\end{equation*}
\normalsize
where $k\ge 2$ is equal to $2^{d}$.
\end{itemize}

\end{corollary}
\begin{proof}
\begin{itemize}

\item[$(a)$] In this case we have that $\mathsf{j}= 2d-2$ and the number of blocks in this case is equal to $d-1$, we also have $t_{d-1}=d, t_d=d-1,\dots ,t_{\sf j}=1$. For each $d\leq i\leq \sf j$ we have that $T_i=\left(1,2,1\right)$, and clearly $A_i=3$ and $S_i=1$. So the total number of partitions of diagonal lengths $T$ is $A=3^{d-1}$. On the other hand, using (\ref{specialmultiblock}), we obtain the number of  special partitions
\begin{align*}
S&=\sum_{i=1}^{d-1}(-1)^{i+1}\sum_{\lambda\subseteq\{d,\dots ,2d-2\}, \vert\lambda\vert=i}1^i \cdot 3^{d-1-i}\\
&=\sum_{i=1}^{d}(-1)^{i+1}\binom{d-2}{i}3^{d-1-i}\\
&=3^{d-1 }-2^{d-1}.
\end{align*}
Thus the number of complete intersection Jordan types with the Hilbert function in $(a)$ is equal to $A-S=2^{d-1}$.
\item[$(b)$] In this case we have that $\mathsf{j}=2d-3+k$ and $t_{d-1}=\cdots =t_{k+d-2}=d$, $t_{k+d-1}=d-1, \dots , t_{2d+k-3}=1$.  For each $i\in [d,d+k-3]$ we have $T_i=0$ and clearly  $A_i=1$ and $S_i=0$. We have $T_{d+k-2}=(1,1)$, so $A_{d+k-2}=2$ and $S_{d+k-2}=0$. There are $d-1$ more components for each $i\in [d+k-1, 2d+k-3]$ where $T_i=(1,2,1)$, $A_i=3$ and $S_i=1$, similar to the previous case. So the total number of  partitions in this case is $A=2\cdot 3^{d-1}$ \par
\noindent Using Equation (\ref{specialmultiblock}) we obtain the number of special partitions
\begin{align*}
S&=\sum_{i=1}^{d+k-2}(-1)^{i+1}\sum_{\lambda\subseteq\{d,\dots ,2d-2\}, \vert\lambda\vert=i}1^i \cdot 3^{d-1-i}\cdot 2\\
&=2\sum_{i=1}^{d-1}(-1)^{i+1}\binom{d+k-3}{i}3^{d-1-i}\\
&=2\cdot 3^{d-1 }-2^{d}.
\end{align*}
Therefore the number of complete intersection Jordan types in this case is equal to $A-S=2^d$.
\end{itemize}
\end{proof}\begin{ack}
The first author was supported by the Swedish Research Council grant
VR 2013-4545. The authors are grateful to the organizers of the conference ``Lefschetz Properties in Algebra, Geometry and Combinatorics'' at  Centro Internazionale per la Ricerca Matematica (CIRM) at Levico, Italy, in June 2018; and to the successor conference ``Lefschetz Properties in Algebra, Geometry and Combinatorics, II'' at
Centre International de Rencontres Math\'{e}matiques (CIRM) at Lumini, France in October 2019, where they participated in the working group on Jordan type. Helpful comments of a referee led us to focus the paper and improve clarity.\end{ack}
\small

config: arxiv