text:
\begin{document}

\begin{abstract}
We prove that the generalized moment-cumulant relations introduced  in
[arXiv:1711.00219] are given by the action of the Eulerian idempotents on the Solomon-Tits algebras,
whose direct sum builds up the Hopf algebra of Word Quasi-Symmetric Functions $\WQSym$.
We prove $t$-analogues of these identities (in which the coefficient of $t$ gives back the original version),
and a similar $t$-analogue of Goldberg's formula for the coefficients of the Hausdorff series. This
amounts to the determination of the action of all the Eulerian idempotents on a product of exponentials.
\end{abstract}

\maketitle

\section{Introduction}

The relation between  moments $m_n=\E(X^n)$ and classical cumulants
$K_n(X)$ of a random variable $X$, encoded in the exponential generating functions
\begin{equation}
  \label{eq:defclass}
\sum_{n\ge0} m_n \frac{t^n}{n!} = \exp\left(\sum_{n\ge 1}K_n \frac{t^n}{n!} \right)
\end{equation}
is, up to a rescaling by factorials, essentially the same as the relation
between complete symmetric functions $h_n$ and the power sum symmetric functions
$p_n$
\begin{equation}
\sum_{n\ge 0}h_n t^n = \exp\left(\sum_{n\ge 1}\frac{p_n}{n} t^n\right).
\end{equation}
Hence we can define a character on the Hopf algebra of symmetric functions $Sym$  by setting
\begin{equation}
\chi_X(h_n) = \frac{1}{n!}\E(X^n).
\end{equation}
Then, the cumulants are given by
\begin{equation}\label{eq:addcum}
K_n(X)=\chi_X\left(\frac{p_n}{(n-1)!}\right)
\end{equation}
and the property that $K_n(X+Y)=K_n(X)+K_n(Y)$ whenever $X$ and $Y$
are independent random variables corresponds in this context
to the fact that the power-sums are primitive elements for the coproduct
\eqref{eq:Sym:coproduct}.

In a more general setting,
with a sequence $(X_i)_{i\ge 1}$ of random variables, one can define multilinear moments
\begin{equation}
m_n := \E(X_1X_2\cdots X_n).
\end{equation}
Stochastic independence can then be algebraically reformulated
in terms of subalgebras of the algebra ${\mathcal X}$
of random variables.
A family $({\mathcal X}_i)_{i\in I}$ of subalgebras of  ${\mathcal X}$
is said to be \emph{independent} if the factorization
\begin{equation}\label{eq:mixmom}
\E(X_1X_2\cdots X_n) = \prod_{B\in\pi}\E\left(\prod_{i\in B}X_i\right)
\end{equation}
for any partition $\pi$ of $[n]$ such that for each block $B$ of $\pi$,
the $X_i$ for $i\in B$ are in the same subalgebra ${\mathcal X}_{j(B)}$
(and $j(B)\not=j(B')$ for $B\not=B'$).
The multivariate classical cumulants are then defined as
\begin{equation}
K_n(X_1,X_2,\dots,X_n) = \left. \frac{\partial}{\partial t_1} \frac{\partial}{\partial
  t_2}\dotsm\frac{\partial}{\partial t_n}\right|_{{\bf t}=0} \IE e^{t_1X_1+t_2X_2+\dots+t_nX_n}
\end{equation}
which coincides with
  \eqref{eq:defclass} when $X_1=X_2=\dotsm=X_n=X$.
These
cumulants are multilinear maps $K_n(X_1,\ldots,X_n)$ whose fundamental property,
generalizing \eqref{eq:addcum}, is that mixed cumulants vanish
in the sense that $K_n(X_1,X_2,\dots,X_n)=0$ whenever
at least two independent subalgebras occur.

In noncommutative probability, the $X_i$ belong to some noncommutative
algebra ${\mathcal A}$, and new notions of independence arise, for which
the factorization of the moments \eqref{eq:mixmom} is replaced
by other identities. Each notion of independence
gives rise to appropriate version of cumulants, such that the vanishing
mixed cumulants (or some weaker condition as in the case of monotone
independence) characterizes independence.

In order to give a unified treatment of all these independences, the notions
of exchangeability system \cite{Leh04} and of spreadability system \cite{HL} have been introduced.
A spreadability system for a noncommutative probability space $({\mathcal A},\varphi)$
allows to produce independent copies $X^{(i)}$ of the random variables, and to formulate
independence in terms of identities satisfied by the moments
$\varphi(X_1^{(i_1)}X_2^{(i_2)}\cdots X_n^{(i_n)})$.
It is in particular assumed that this quantity depends only on the \emph{packed word}
$u=\pack(i_1i_2\cdots i_n)$ (or \emph{ordered set partition} \cite{HL}).
This implies that for a spreadability  system, any choice of
a sequence $(X_i)$ of random variables determines a linear form on an appropriate Hopf
algebra based on packed words, namely, $\WQSym^*$, the graded dual of Word Quasi-Symmetric functions,
also known as quasi-symmetric functions in noncommuting variables \cite{NT06,BZ,NCSF7}.
$\WQSym$ is a noncommutative version of the algebra of quasi-symmetric functions.
As shown by Hivert \cite{Hiv}, the quasi-symmetric polynomials are actually the invariants of
a very peculiar action of the symmetric group $\SG_n$ on polynomials in $n$ variables, called the quasi-symmetrizing
action. This action can be extended to polynomials in noncommuting variables \cite{NCSF7}, and letting $n$ tend to infinity,
the algebra of invariants acquires a Hopf algebra structure, just as in the case of symmetric polynomials.

It turns out that the homogenous component of degree $n$ of its graded dual, $\WQSym_n^*$, can be identified
with the Solomon-Tits algebra of $\SG_n$, and that this identification is compatible with that
of $\Sym_n$, the space of noncommutative symmetric functions of degree $n$, with the ordinary Solomon descent algebra.
That is, there is an embedding of Hopf algebras $\Sym\hookrightarrow \WQSym^*$ which is compatible with the internal products.

The moment-cumulant relations of \cite{HL} actually describe  the relation between the natural basis
$\N_u$ of $\WQSym^*$ and its internal product with the  Eulerian idempotents of the descent algebra $\Sym_n$.

We provide simple conceptual proofs of most of the identities of \cite{HL}, and give a complete
description of the action of the Eulerian idempotents on $\WQSym^*$. As shown in \cite{FPT},
the celebrated Goldberg formula for the coefficients of the Hausdorff series
\begin{equation}
H(a_1,a_2,\ldots) = \log(e^{a_1}e^{a_2}\cdots) = \sum_{w\in A^*}c_w w
\end{equation}
amounts to a description
of the action of the first Eulerian idempotent on $\WQSym^*$. Our results allow us to
compute the coefficients of the expansion
\begin{equation}
(e^{a_1}e^{a_2}\cdots)^t = \sum_{w\in A^*}c_w(t) w
\end{equation}
in which the Hausdorff series is the coefficient of $t$.

\section{Background}

In this section, we recall the basic definitions of the various Hopf algebras involved in the sequel:
ordinary ($Sym$) and noncommutative ($\Sym$) symmetric functions, word quasi-symmetric functions
($\WQSym$) and word symmetric functions ($\WSym$).

Given a sequence $(X_i)$ of random variables, the moments
$\varphi(X_1\cdots X_n)$ determine a character of $\Sym$. We shall see that the
the structures introduced in \cite{Leh04,HL} allow to extend this character to a linear map
of $\WQSym$ (in the case of spreadability systems) or  $\WSym$ (in the case of exchangeability systems).

\subsection {Ordinary symmetric functions}
Let $X=\{x_i|i\ge 1\}$ be an infinite set of commuting variables.
The \emph{complete homogeneous functions} $h_n(X)$ and the
\emph{elementary symmetric functions} $e_n(X)$ are
\begin{equation}
h_n(X)=\sum_{i_1\le i_2\le \dots\le i_n}x_{i_1}x_{i_2} \dots x_{i_n}
\end{equation}
that is, is the sum of all monomials of total degree $n$, and
\begin{equation}
e_n(X)=\sum_{i_1<i_2<\dots<i_n}x_{i_n}x_{i_{n-1}} \dots x_{i_1}
\end{equation}
is the sum of all products of $n$ distinct variables.
These functions are invariant under permutation of the variables
and both sequences freely generate the algebra $Sym(X)$ of
\emph{symmetric functions}
\begin{equation}
Sym = \KK[h_1,h_2,\ldots] = \KK[e_1,e_2,\ldots].
\end{equation}
We can thus define a coproduct
\begin{equation}
  \label{eq:Sym:coproduct}
\Delta h_n = \sum_{k=0}^nh_k\otimes h_{n-k}
\end{equation}
which endows it with the structure of a graded Hopf algebra.
Given a second alphabet $Y$ disjoint from $X$, and identifying a tensor product
$f\otimes g$ with $f(X)g(Y)$, the coproduct is given by $\Delta(f)=f(X+Y)$,
where by $X+Y$ we denote the (still countable) union of the alphabets
$X$ and $Y$.
It can be shown that $Sym$ is self-dual.
 Its primitive elements
are the power-sums $p_n=\sum_i x_i^n$, which also generate $Sym$ over the rationals.

For a partition $\mu=(1^{m_1}2^{m_2}\cdots n^{m_n})$ of $n$, define
\begin{equation}
p_\mu = \prod_{i=1}^n p_i^{m_i}\quad\text{and}\quad z_\mu=\prod_{i=1}^n i^{m_i}m_i!.
\end{equation}
Each homogeneous component $Sym_n$ is endowed with a unique \emph{internal
  product}
$*$
by declaring  the elements $\frac{p_\mu}{z_\mu}$ to be orthogonal idempotents.

\subsection{Classical cumulants}

The classical cumulants $K_n$ of a random variable $X$ are related to its
moments $m_n=E[X^n]$ by
\begin{equation}
\sum_{n\ge 0}m_n\frac{t^n}{n!} = \exp\left(\sum_{n\ge 1}K_n\frac{t^n}{n!}\right)
\end{equation}
which differs from the relations between complete homogeneous symmetric functions $h_n$ and
power-sums $p_n$ by a simple rescaling:
\begin{equation}
\sum_{n\ge 0}h_n t^n = \exp\left(\sum_{n\ge 1}\frac{p_n}{n} t^n\right).
\end{equation}

Identifying $m_n$ with $n!h_n$, the cumulants $K_n$ become identified with $(n-1)!p_n$.
A random variable $X$ determines a character $\chi_X$ of the algebra of symmetric
functions, by defining
\begin{equation}
\chi_X(h_n) = \frac{m_n}{n!}\ \Longleftrightarrow\ \chi_X(p_n)=\frac{K_n}{(n-1)!}.
\end{equation}
A convenient symbolic notation for characters of $Sym$ is that of virtual alphabets:
given a sequence of algebraic generators $g_n$ of $Sym$ (such as $h_n,p_n,e_n,\ldots$),
one denotes $\chi(g_n)$ by $g_n(\XX)$, where the symbol $\XX$ is called the virtual
alphabet associated with $\chi$. Then, if $\eta$ is another character whose virtual alphabet
is $\YY$, the convolution $\chi\star\eta := (\chi\otimes\eta)\circ\Delta$ is given
by
\begin{equation}
(\chi\star\eta)(f)=f(\XX+\YY).
\end{equation}
On can thus associate with each random variable $X$ a virtual alphabet $\XX$,
such that $m_n(X)=n!h_n(\XX)$. If $X$ and $Y$ are independent, then

\begin{equation}
\chi_{X+Y}(h_n)=\frac{m_n(X+Y)}{n!}=h_n(\XX+\YY)= (\chi_X\otimes\chi_Y)\Delta h_n,
\end{equation}
that is, $\chi_{X+Y}$ is the convolution of  $\chi_X$ and $\chi_Y$.
The additivity of cumulants on independent variables corresponds to  the fact
that the power-sums are primitive elements.
This can be seen as another incarnation of the Hopf-algebraic
version of the \emph{umbral calculus} \cite{JoniRota:coalgebras}.

\begin{example}[The formula of
  Good and Cartier]
Let $Y=\Omega=\{1,\omega,\ldots,\omega^{n-1}\}$ be the alphabet of $n$-th roots of unity.
These are the roots of the polynomial $x^n-1$.
Using the factorization $x^n-1=(x-1)(1+x+\dots+x^{n-1})$ we see that all the roots $\omega^k$
for $1\leq k\leq n-1$  are roots of the second factor and therefore
$p_k(\Omega)=0$ for $1\leq k\leq n-1$, whereas $p_n(\Omega)=n$.
Thus we have
\begin{equation}
h_n(\Omega X)=
\sum_{\lambda\vdash n}\frac{p_\lambda(\Omega) \, p_\lambda(X)}{z_\lambda}
= p_n(X).
\end{equation}

Identifying as above $m_n$ with $n!h_n$, $K_n$ becomes identified with
$(n-1)!p_n$, and we obtain the formula

\begin{equation}\label{eq:omegacom}
nK_n = \E[(X^{(1)} + \omega X^{(2)} +\cdots +\omega^{(n-1)}X^{(n)})^n],
\end{equation}
known as \emph{Good's formula} in the mathematics literature  \cite{Good:1975:new}
and \emph{Cartier's formula} for Ursell functions in the physics literature
\cite{Percus:1975:correlation,Simon:1993:statistical}.

\end{example}

\subsection{Noncommutative symmetric functions}
Let $A=\{a_i|i\ge 1\}$ be an infinite totally ordered set of noncommuting variables.
We set
\begin{equation}
\label{Sn-G-fqs}
S_n(A) = \sum_{i_1\le i_2\le \dots\le i_n}a_{i_1}a_{i_2} \dots a_{i_n}
\end{equation}
and
\begin{equation}
\Lambda_n(A) = \sum_{i_1<i_2<\dots<i_n}a_{i_n}a_{i_{n-1}} \dots a_{i_1}
\end{equation}
and call them respectively \emph{noncommutative complete functions} and
\emph{noncommutative elementary function}s of $A$.

The $S_n(A)$ freely generate a subalgebra of the formal power series over $A$ which is denoted by
$\Sym(A)$ and called \emph{noncommutative symmetric functions} \cite{NCSF1}.

The $S_n(A)$ and $\Lambda_n(A)$ have the simple noncommutative generating
series
\begin{equation}
\sigma_t(A) := \sum_{n\ge 0}t^n S_n(A)
             = \prod_{i\ge 1}^\rightarrow (1-ta_i)^{-1}
\end{equation}
and
\begin{equation}
\label{lambdaNC}
\lambda_{-t}(A) := \sum_{n\ge 0}(-t)^n\Lambda_n(A)
                 = \prod_{i\ge 1}^\leftarrow(1-ta_i)=\sigma_t(A)^{-1}
\end{equation}
where we have set $S_0=\Lambda_0=1$, and $t$ is an indeterminate commuting
with $A$.
We shall almost always forget about the alphabet $A$ since the context is
generally clear.

From the generators $S_n$, we can form a linear basis
\begin{equation}
S^I = S_{i_1}S_{i_2}\cdots S_{i_r}
\end{equation}
of the homogeneous component $\Sym_n$, parametrized by
integer compositions of $n$.
The dimension of $\Sym_n$ is thus $2^{n-1}$ for $n\ge 1$.

The coproduct
\begin{equation}
\Delta S_n = \sum_{k=0}^n S_k\otimes S_{n-k}
\end{equation}
endows $\Sym$ with the structure of a graded Hopf algebra. Since this coproduct
is clearly cocommutative, $\Sym$ cannot be self-dual. Its graded dual is $QSym$,
the algebra of quasi-symmetric functions \cite{NCSF1}. The dual basis of $S^I$ is the quasi-monomial
basis $M_I$ of $QSym$.

The map $S_n\mapsto h_n$ is a morphism of Hopf algebras $\Sym\rightarrow Sym$,
dual to the natural embedding of $Sym$ into $QSym$.

Each homogeneous component $\Sym_n$ is endowed with an internal product $*$
for which $\Sym_n$ is anti-isomophic to the Solomon descent algebra $\Sigma_n$
of $\SG_n$. The basis element $S^I$ is identified with the formal sum
of all permutations whose descent composition is coarser than $I$.

Finally, the primitive elements of $\Sym$ span a free Lie algebra, generated by
the $\Phi_n$ defined by
\begin{equation}
\sum_{n\ge 1}\frac{\Phi_n}{n} =\log\left(\sum_{n\ge 0}S_n\right).
\end{equation}
The $\varphi_n:=\frac1n\Phi_n$ are idempotents for the internal product. These
are the Solomon idempotents, also called (first) Eulerian idempotents.

\subsection{Cumulants in noncommutative probability}

A noncommutative probability space is a pair $(\AA,\varphi)$ where $\AA$ in a unital
algebra and $\varphi$ a linear form, (or more generally a linear map $\AA\rightarrow \BB$
for some algebra $\BB$ such that $\AA$ is a $\BB$-module) such that $\varphi(1)=1$.

Elements of $\AA$ are called random variables. To define moments in this context,
we need an infinite sequence $(X_i)_{i\ge 1}$ of random variables. Then, the  moments are
\begin{equation}
m_n := \varphi(X_1X_2\cdots X_n).
\end{equation}
Such a sequence determines a character $\hat\varphi$ of $\Sym$ by setting
\begin{equation}
\hat\varphi(S_n)=m_n
\end{equation}
(we drop the factorials which are irrelevant in this context).

Cumulants are defined with respect to a notion of independence, leading to several
notions such as free, monotone, or boolean cumulants, and many others.

Attempts to
give a unified treatment of all these notions have led to the introduction \cite{Leh04} of
the notion of exchangeability systems, and later  \cite{HL} of
the more general notion of spreadability systems. The interpretation of this formalism
in terms of combinatorial Hopf algebras requires the introduction of $\WQSym$ (Word Quasi-Symmetric functions)
and its dual $\WQSym^*$. A spreadability system determines an extension of the character $\hat\varphi$
to $\WQSym^*$, and the various notions of independence reflect certain symmetries of this extension.

\subsection{Word quasi-symmetric functions}

\def\PW{{\rm PW}}
\def\NCQSym{{\bf NCQSym}}

The \emph{packed word} $u=\pack(w)$ associated with a word $w\in A^*$ is
obtained by the following process. If $b_1<b_2<\ldots <b_r$ are the letters
occuring in $w$, $u$ is the image of $w$ by the homomorphism
$b_i\mapsto a_i$. We usually represent  $a_i$ by $i$ in the indexation
of bases.

A word $u$ is said to be \emph{packed} if $\pack(u)=u$. We denote by $\PW$ the
set of packed words.
With such a word, we associate the ``polynomial''
\begin{equation}
\M_u :=\sum_{\pack(w)=u}w\,.
\end{equation}

For example, restricting $A$ to the first five integers,
\begin{equation}
\begin{split}
\M_{13132}=&\ \ \ 13132 + 14142 + 14143 + 24243 \\
&+ 15152 + 15153 + 25253 + 15154 + 25254 + 35354.
\end{split}
\end{equation}

For a word $w\in A^*$ and a letter $a\in A$ we denote by $\abs{w}_a$ the number of occurences
of the $a$ in $w$.
The \emph{evaluation} $\ev(w)$  is then the vector obtained
from the sequence $(\abs{w}_a)_{a\in A}$ by removing all zeros.

Under the abelianization
$\chi:\ \KK\langle A\rangle\rightarrow\KK[X]$, the $\M_u$ are mapped to the monomial
quasi-symmetric functions
\begin{equation}
M_I := \sum_{j_1<j_2<\dots<j_r}x_{j_1}^{i_1}x_{j_2}^{i_2}\cdots x_{j_r}^{i_r},
\end{equation}
where $\ev(u)=(i_1,\ldots,i_r)$ is the evaluation vector of $u$.

The $\M_u$ span a subalgebra
of $\K\langle A\rangle$, called $\WQSym$ for Word
Quasi-Symmetric functions,
consisting in the invariants of the noncommutative
version of Hivert's quasi-symmetrizing action \cite{Hiv,NCSF7}.

Packed words can be naturally identified with \emph{ordered set partitions},
the letter $a_i$ at the $j$th position meaning that $j$ belongs to block $i$.
For example,
\begin{equation}

u=313144132 \ \leftrightarrow\ \Pi=(\{2,4,7\},\{9\},\{1,3,8\},\{5,6\})\,.
\end{equation}

Let $\N_u\in\WQSym^*$ be the dual basis of $\M_u$. Define an internal product
on $\WQSym^*_n$ by \cite{NT06}
\begin{equation}
\N_u * \N_v = \N_{\pack{u\choose v}},
\end{equation}
where ${u\choose v}$ denotes the word in {\it biletters} ${u_i\choose v_i}$, lexicographically ordered
with priority to the top letter.
For example,

$$
\pack
\binom{1\,2\,1\,1\,3\,1}{2\,2\,1\,3\,1\,1}

=241351
$$

Then,
\begin{proposition}
$(\WQSym^*,*)$ is anti-isomorphic to the Solomon-Tits algebra.
\end{proposition}
Indeed, if one writes $u=\{s'_1,\ldots,s'_k\}$ and $v=\{s''_1,\ldots,s''_l\}$ as
ordered set partitions, then the packed word $w=\pack{u\choose v}$
corresponds to the ordered set partition obtained from
\begin{equation}
\{ s'_1\cap s''_1, s'_1\cap s''_2, \ldots, s'_1\cap s''_l,
   s'_2\cap s''_1, \ldots, s'_k\cap s''_l
   \}.
\end{equation}

Finally,
\begin{equation}
S^I = \sum_{\ev(u)=I}\N_u
\end{equation}
defines an embedding of Hopf algebras compatible with the internal product $*$, that is, inducing
the standard embedding of the descent algebra into the Solomon-Tits algebra.

\subsection{Spreadability systems and exchangeability systems}

A spreadability system for $(\AA,\varphi)$ is a triple $(\UU,\tilde\varphi,(I^{(i)}))$,
where $(\UU,\tilde\varphi)$ is a $\BB$-valued noncommutative probablility space,
$I^{(i)}$ is a morphism $\AA\rightarrow \UU$ such that $\varphi=\tilde\varphi\circ I^{(i)}$
for all $i$, and, setting $X^{(j)}:=I^{(j)}(X)$,
\begin{equation}
\tilde\varphi(X_1^{(i_1)}X_2^{(i_2)}\cdots X_n^{(i_n)})
=
\tilde\varphi(X_1^{(j_1)}X_2^{(j_2)}\cdots X_n^{(j_n)})
\end{equation}
whenever $\pack(i_1\cdots i_n)=\pack(j_1\cdots j_n)$.

Thus, for each sequence $(X_i)_{i\ge 1}$ of random variables in $\AA$, a spreadability
system determines a linear map $\hat\varphi:\ \WQSym^*\rightarrow \BB$ by
\begin{equation}
\hat\varphi(\N_u) := \varphi(X_1^{(u_1)}\cdots X_n^{(u_n)}).
\end{equation}
In \cite{HL}, this is denoted by $\varphi_\pi(X_1\cdots X_n)$, where $\pi$
is the ordered set partition encoded by the packed word $u$.

The notion of ${\mathcal S}$-independence \cite{HL} can be reformulated in terms of the internal product
of $\WQSym^*$. A family of subalgebras $\AA_i$ is ${\mathcal S}$-independent
if, when $X_j\in \AA_{v_j}$,
\begin{equation}\label{eq:Sindep}
\hat\varphi(\N_u)=\hat\varphi(\N_u*\N_v).
\end{equation}

An exchangeability system is a spreadability system satisfying
\begin{equation}
\hat\varphi(\N_{\sigma(u)}) = \hat\varphi(\N_u)).
\end{equation}
for all permutations $\sigma$ of the alphabet of $u$. In this case,
$\hat\varphi$ can be interpreted as a character of $\WSym^*$,
an algebra based on set partitions, to be defined below.

\subsection{Symmetric functions in noncommuting variables}

Let $A$ be
an alphabet, then every permutation $\sigma\in\SG(A)$ gives rise
to an automorphism
on the free algebra $\K\<A\>$. Two words $u=u_1\cdots u_n$ and $v=v_1\cdots v_n$
are in the same orbit whenever $u_i=u_j \Leftrightarrow v_i=v_j$. Thus, the
orbits are in one-to-one correspondence with set partitions into at most $|A|$ blocks. Assuming as above
that $A$ is infinite, we obtain an algebra based on all set partitions,
defining  the monomial basis by
\begin{equation}
\m_\pi(A)=\sum_{w\in O_\pi}w
\end{equation}
where $O_\pi$ is the set of words such that $w_i=w_j$ iff $i$ and $j$ are
in the same block of $\pi$.

One can introduce a bialgebra structure by means of the coproduct
\begin{equation}
\Delta f(A) = f(A'+A'')
\end{equation}
where $A'+A''$ denotes the disjoint union of two mutually commuting copies of
$A$. Again,
the coproduct of a monomial function is clearly
\begin{equation}
\Delta \m_\pi = \sum_{\pi'\vee\pi''=\pi}\m_{\std(\pi')}\otimes \m_{\std(\pi'')}
\end{equation}
This coproduct is obviously cocommutative.

Let $\le$ be the {\it reverse} refinement order on set partitions
($\pi\le\pi'$ means that $\pi$ is coarser than $\pi'$, i.e. that its
blocks are union of blocks of $\pi'$)\footnote{We need this reverse order to
be compatible with the usual conventions for symmetric functions. We reserve
the notation $\preceq$ for the usual order on set partitions.}.

The basis $\Phi^\pi$, defined by sums over initial intervals
\begin{equation}
\Phi^\pi =\sum_{\pi'\le\pi}\m_{\pi'}
\end{equation}
is multiplicative with respect to concatenation of set partitions
and hence $\WSym$ is freely generated by the elements $\Phi^\pi$ such that
$\pi$ is irreducible, i.e.,
the coarsest interval partition dominating $\pi$ has only one block.

The graded dual of $\WSym$ is a commutative
algebra, isomorphic to the algebra  $\Pi QSym$
 defined in
\cite[Sec 3.5.1]{HNT08}.
Let $N_\pi$ be the dual basis of $\m_\pi$ in the (commutative) graded  dual
$\WSym^*\simeq \Pi QSym$
and let $\phi_\pi$ be the dual basis of $\Phi^\pi$.
Then,
\begin{equation}\label{eq:Nphi}
N_{\pi'} =\sum_{\pi\geq\pi'}\phi_{\pi}.
\end{equation}
$\WSym^*$ is a quotient of $\WQSym^*$, defined by $\N_u\equiv \N_v$ iff $u=\sigma(v)$ for some permutation
$\sigma$ of the alphabet of $u$ (e.g., $\N_{121}\equiv \N_{212}$). Equivalence classes are parametrized by
the set partitions corresponding to any of the set compositions encoded by equivalent packed words.
The internal product of $\WQSym^*$ descends to an internal product on $\WSym^*$, which is given by the meet
of the lattice of set partitions:
\begin{equation}
N_\pi*N_\tau = N_{\pi\wedge\tau}
\end{equation}
where $\pi\wedge\tau$ is the coarsest partition which is finer than $\pi$ and $\tau$.

For an echangeability system, the notion of ${\mathcal E}$-independence translates as
\begin{equation}
\hat\varphi(N_\pi)=\hat\varphi(N_\pi*N_\tau)
\end{equation}
under the same hypotheses as in \eqref{eq:Sindep}.

Endowed with this product, the homogenous component $\WSym_n^*$ is known as the Moebius algebra
of the partition lattice $\Pi_n$ \cite{Sol67,Gre73}. It has been shown by Solomon that  $\{\phi_\pi|\pi\in\Pi_n\}$
is a complete set of orthogonal idempotents of $\WSym^*_n$.

As a consequence, if $\pi$ is not the trivial partition $\{12\ldots n\}$,
\begin{equation}\label{eq:prodNphi}
N_\pi*\phi_{\{12\cdots n\}}=0.
\end{equation}

\subsection{Noncommutative version of Good's formula}

Although \eqref{eq:omegacom} is essentially trivial, it has a not-so-trivial noncommutative
analogue in noncommutative symmetric functions. It is proved in \cite[Prop. 8.6]{NCSF2}
that
\begin{equation}
S_n(\Omega A) = K_n(\omega)
\end{equation}
where $K_n(\omega)$ is Klyachko's element, a famous Lie (quasi-) idempotent.

This fact leads to an interesting interpretation of \cite[Definition 2.1]{Leh04}.
To understand it, we have to follow the chain of morphisms
\begin{equation}
\Sym\hookrightarrow \WQSym^* \twoheadrightarrow \WSym^* \simeq {\Pi}Qsym.
\end{equation}

The first embedding $i$ is given by
\begin{equation}
i:\ S^I \mapsto \sum_{\ev(u)=I}\N_u
\end{equation}
where $\N_u=\M_u^{*}$. The projection $p$ onto $\WSym^*$ is dual to the inclusion  of $\WSym$
into $\WQSym$, and therefore given by $\N_u \mapsto N_\pi$, where $N_\pi$ is dual to the monomial
basis of $\WSym$, and $\pi$ is the set partition underlying the set composition encoded by $u$.

In \cite{Leh04},  cumulants for an exchangeability system are defined by a noncommutative analogue of Good's formula.
We shall see that they can
be interpreted as the image of $(n-1)!K_n(\omega)$ under the composition $\xi=p\circ i$ of these two maps.

Since $\xi$ is valued in a commutative algebra, it factors through $Sym(X)$, and all noncommutative
power sums have the same image. The choice of Klyachko's element is therefore arbitrary.

It remains to determine its image. Let $\Phi^\pi = \sum_{\pi\le\tau}\M_\tau$ be the power-sum basis
of $\WSym$, and let $\phi_\pi$ be its dual basis. Under the commutative image map
\begin{equation}
\chi:\ \WSym(A) \twoheadrightarrow Sym(X)
\end{equation}
given by $f(A)\mapsto f(X)$, we have
\begin{equation}
\M_\pi(A) \mapsto \prod_i m_i(\lambda)! \cdot m_\lambda(X)
\end{equation}
where $\lambda$ is the integer partition associated with $\pi$,
$m_\lambda(X)$ is the monomial symmetric function,  $m_i( \lambda)$ is the
multiplicity of the part $i$ in $\lambda$, and
\begin{equation}
\Phi^\pi(A) \mapsto  p_\lambda(X).
\end{equation}
Dually,
\begin{equation}
\chi^*(h_\lambda) = \prod_i m_i(\lambda)! \sum_{\Lambda(\pi)=\lambda}N_\pi,
\end{equation}
and
\begin{equation}
\chi^*(p_\lambda) = z_\lambda \sum_{\Lambda(\pi)=\lambda}\phi_\pi.
\end{equation}
The image of  any Lie idempotent of $\Sym$ in $\WSym^*$ is therefore the same as that
of $\varphi_n = \frac1n\Phi_n$, which is
\begin{equation}
\varphi_n =
\sum_{I\vDash n} \frac{(-1)^{\ell(I)-1}}{\ell(I)}S^I
\mapsto    \sum_{|u|=n}\frac{(-1)^{\max(u)-1}}{\max(u)}\N_u
\mapsto
\sum_{\pi\vdash [n]}\frac{(-1)^{\ell(\pi)-1}}{\ell(\pi)}\ell(\pi)!N_\pi.
\end{equation}
Since this must also be equal to
\begin{equation}
\chi^*(\frac1n p_n) =\phi_{\{1,2,\ldots, n \} },
\end{equation}
and since, by definition of the Moebius function of the lattice of partitions
\begin{equation}
\phi_{\{1,2,\ldots, n \} }=\sum_{\pi\vdash [n]}\mu(\pi,\hat 1_n)N_\pi
\end{equation}
we recover the classical fact that
\begin{equation}
  \label{eq:moebiusmu}
\mu(\pi,\hat 1_n)=(-1)^{\ell(\pi)-1}(\ell(\pi)-1)!
\end{equation}
by merely contemplating a chain of morphisms.

\subsection{Cumulants for exchangeability systems}

An exchangeability system together with a sequence $(X_i)_{i\ge 1}$ determines
a linear form $\hat\varphi$ on $\WSym^*$
\begin{equation}
\hat\varphi(N_\pi) := \varphi(X_1^{(u_1)}\cdots X_n^{(u_n)}).
\end{equation}
where $u$ is any packed word representing the set partition $\pi$.

The partitioned moments are
\begin{equation}
\varphi_\pi(X_1\cdots X_n) = \hat\varphi(N_\pi),
\end{equation}
and the cumulants defined in \cite{Leh04} are
\begin{equation}
K_\pi(X_1,\ldots,X_n) =\hat\varphi(\phi_\pi).
\end{equation}
By \eqref{eq:Nphi}, we have \cite[Prop. 2.7]{Leh04}
\begin{equation}
  \label{eq:phipi=sumksig}
\varphi_\pi(X_1,\ldots,X_n)=\sum_{\pi \le\sigma}K_\sigma(X_1,\ldots,X_n).
\end{equation}
The  independence condition  defined by \eqref{eq:Sindep} becomes, for
exchangeability systems
\begin{equation}
\hat\varphi(N_\pi)=\hat\varphi(N_\pi*N_\sigma)=\hat\varphi(N_{\pi\wedge\sigma}).
\end{equation}
This implies the vanishing of mixed cumulants:
if $\pi$ is a partition if $[n]$
into two nontrivial blocks $b_1,b_2$ such that $\{X_i|i\in b_1\}$ and $\{X_i|i\in b_2\}$
are independent, then the cumulant
\begin{equation}
K_n(X_1,\ldots,X_n)=\hat\varphi(\phi_{1^n})=\hat\varphi(N_\pi*\phi_{1^n}) = 0
\end{equation}
by \eqref{eq:prodNphi}.

Monotone independence is not a special case of ${\mathcal E}$-independence, but can be recovered
from the notion of
${\mathcal S}$-independence. In this case, the requirement that mixed cumulants vanish is too
strong, and is replaced by the condition that they should be eigenfunctions of Rota's dot
multiplication by and integer:
\begin{equation}
K_u(N.X_1,\ldots,N.X_n)=N^{\max(u)}K_u(X_1,\ldots,X_n)
\end{equation}
where $u$ is a packed word, and $N.X=X^{(1)}+\cdots+X^{(N)}$ is the sum of
$N$ independent copies of $X$.

We shall see that the dot operation translates as multiplication of the alphabet by $N$
in the relevant Hopf algebras. The operator $f(A)\mapsto f(NA)$ is semisimple, and its spectral
projectors are know as the Eulerian idempotents.

\section{Review of the Eulerian algebra}

\subsection{Basics}
The Eulerian algebra is a commutative subalgebra of dimension $n$ of the group algebra of
the symmetric group $\SG_n$, and in fact of its descent algebra $\Sigma_n$. It was apparently
first  introduced in \cite{BMP} under the name {\em algebra of permutors}\footnote{
A self-contained and elementary presentation of the main results of \cite{BMP}
can be found in \cite{NCSF1}.}.
It is spanned by the Eulerian idempotents, or, equivalently, by the sums of
permutations having the same number of descents.

It is easier to work with all symmetric groups at the same time, with the
help of generating functions. Recall that the algebra of noncommutative
symmetric functions $\Sym$ is endowed with an internal product $*$,
for which each homogeneous component $\Sym_n$ is anti-isomorphic to $\Sigma_n$
\cite[Section 5.1]{NCSF1}.

The {\em Eulerian idempotents} $E_n^{[k]}$ are the homogenous components
of degree $n$ in the series $E^{[k]}$ defined by
\begin{equation}
\sigma_t(A)^x=\sum_{k\ge 0}x^k E^{[k]}(A),
\end{equation}
(see \cite[Section 5.3]{NCSF1}).
We have
\begin{equation}
E_n^{[k]}*E_n^{[l]} = \delta_{kl}E_n^{[k]}\,,\quad\text{and}\quad\sum_{k=1}^nE_n^{[k]}=S_n,
\end{equation}
so that the $E_n^{[k]}$ span a commutative $n$-dimensional $*$-subalgebra of
$\Sym_n$, denoted by $\Eul_n$ and called the Eulerian subalgebra.

\subsection{Noncommutative Eulerian polynomials}

The {\em noncommutative Eulerian polynomials} are defined by \cite[Section 5.4]{NCSF1}

\begin{equation}
{\mathcal A}_n(t) =
\ \sum_{k=1}^n \ t^k\, \Big(
\sum_{{\scriptstyle |I|=n}\atop{\scriptstyle \ell(I)=k}} R_I \,
\Big)
=
\ \sum_{k=1}^n \ {\bf A}(n,k)\, t^k \, ,
\end{equation}

where $R_I$ is the ribbon basis \cite[Section 3.2]{NCSF1}
The following facts can be found (up to a few misprints\footnote{
Eqs. (93) and (97) of \cite{NCSF1} should be read as (\ref{A2S}) and (\ref{Wor}) of the present paper.
}) in \cite{NCSF1}.
The generating series of the ${\mathcal A}_n(t)$ is

\begin{equation}
{\mathcal A}(t) := \ \sum_{n\ge 0} \, {\mathcal A}_n(t)
=
(1-t) \, \left( 1 - t\, \sigma_{1-t} \right)^{-1} \,,
\end{equation}
where $\sigma_{1-t}=\sum (1-t)^nS_n$.

\medskip
Let ${\mathcal A}_n^*(t) = (1-t)^{-n}\, {\mathcal A}_n(t)$.
Then,
\begin{equation}
{\mathcal A}^*(t)
:=
\ \sum_{n\ge 0} \, {\mathcal A}_n^*(t)
=
\sum_{I} \
\left( \displaystyle {t \over 1-t} \right)^{\ell(I)} \, S^I \ .
\end{equation}

This last formula can also be written in the form

\begin{equation} \label{GEN*}
{\mathcal A}^*(t)
=
\ \sum_{k\ge 0} \ \left(
{t\over 1-t}\right)^k \left( S_1+S_2+S_3+\cdots\, \right)^k
\end{equation}

or

\begin{equation}\label{GEN_A}
{1\over 1-t\, \sigma_1(A)}
=
\ \sum_{n\ge 0}\ {{\mathcal A}_n(t)\over (1-t)^{n+1}} \ .
\end{equation}
Let $S^{[k]}=\sigma_1(A)^k$ be the coefficient of $t^k$ in this series. In degree $n$,
\begin{equation}\label{S2E}
S_n^{[k]}=\sum_{I\vDash n, \ell(I)\le k}{k\choose \ell(I)}S^I=\sum_{i=1}^nk^iE_n^{[i]}\,.
\end{equation}
This is another basis of $\Eul_n$.
Expanding the factors $(1-t)^{-(n+1)}$ in the right-hand side
of (\ref{GEN_A}) by the binomial
theorem, and taking the coefficient of $t^k$ in the term of
weight $n$ in both sides, we get

\begin{equation}
S_n^{[k]}
=
\ \sum_{i=0}^k \ {n+i \choose i} \, {\bf A}(n,k-i) \, .
\end{equation}

Conversely,

\begin{equation}
{ {\mathcal A}_n(t) \over (1-t)^{n+1} }
=
\ \sum_{k\ge 0}\ t^k\, S_n^{[k]} \ ,
\end{equation}

so that

\begin{equation}\label{A2S}
{\bf A}(n,p)
=
\ \sum_{i=0}^p\ (-1)^i\, {n+1\choose i}\, S_n^{[p-i]} \ .
\end{equation}

The expansion of the $E_n^{[k]}$
on the basis ${\bf A}(n,i)$, which is a noncommutative analog
of Worpitzky's identity (see \cite{Ga} or \cite{Lod89}) is

\begin{equation}\label{Wor}
\sum_{k=1}^n \ x^k\, E_n^{[k]}
=
\ \sum_{i=1}^n \ {x+n-i\choose n}\, {\bf A}(n,i) \ .
\end{equation}
Indeed, when $x$ is a positive integer $N$,
\begin{equation}
\sum_{k=1}^n \ N^k\, E_n^{[k]}
= S_n(NA) = \sum_{I\vDash n}F_I(N)R_I(A)
\end{equation}
where $F_I$ are the fundamental quasi-symmetric functions,
and for a composition $I=(i_1,\ldots,i_r)$ of $n$,
\begin{equation}
F_I(N)={N+n-r\choose n}\,.
\end{equation}

\section{Adams operations of $\Sym$ and their substitutes}

On any bialgebra $\H$ with multiplication $\mu$ and comultiplication $\Delta$, one
can define the {\it Adams operations}
\begin{equation}
\Psi^k(x) = \mu_k\circ \Delta^k(x)
\end{equation}
where $\Delta^k$ is the iterated coproduct with values in $\H^{\otimes k}$ and $\mu_k$
the multiplication map $\H^{\otimes k}\rightarrow \H$. In other terms, $\Psi^k$ is the
$k$th convolution power of the identity.

On ordinary symmetric functions, these
operations act by multiplication by $k$ of the alphabet, and are therefore algebra
morphisms. However, on noncommutative symmetric functions, the $\Psi^k$ are not anymore
multiplicative, and therefore of lesser interest.

One can replace them by the algebra morphism $\psi_k:\ f(A)\mapsto f(kA)$,
which is diagonalized by the Eulerian idempotents.

Recall that, by definition,
\begin{equation}
\sigma_1(kA) = \sigma_1(A)^k,
\end{equation}
and that for any $f\in\Sym$,
\begin{equation}
f(kA) = f(A) * \sigma_1(kA) = f(A)*\sum_{r\ge 1} k^r E^{[r]}.
\end{equation}
Thus, a simultaneous eigenbasis of the $\psi_k$ is for example
\begin{equation}
K_I := S^I * E^{[\ell(I)]}
\end{equation}
which satisfy
\begin{equation}
K_I(kA) = k^{\ell(I)}K_I(A).
\end{equation}
The basis $K_I$ is actually multiplicative: if $I=(i_1,\ldots,i_r)$,
\begin{align}\label{eq:mult}
K_I& = [k^r] S^I * \sigma_1(A)^k = [k^r]\mu_r[(S_{i_1}\otimes\cdots\otimes S_{i_r})*_r (\sigma_1(kA)\otimes\cdots\otimes \sigma_1(kA)]\\
&= (S_{i_1}*E^{[1]})\cdots  (S_{i_r}*E^{[1]}) = K_{i_1}\cdots K_{i_r}.
\end{align}
Of course, $S_i*E_i^{[1]}=E_i^{[1]}=\frac1i \Phi_i$, so that $K_I$ is just a scaled version of
the classical basis $\Phi^I$.

Since
\begin{equation}
\sigma_1(kA) = [1+(S_1+S_2+S_3+\cdots)]^k = \sum_{I}{k\choose \ell(I)}S^I,
\end{equation}
we have the simple closed form
\begin{equation}
S^I(kA) = \sum_{J\ge I}\beta_k(J,I) S^J,
\end{equation}
where
\begin{equation}
\beta_k(J,I) := \prod_p{k\choose \ell(J_p)},
\end{equation}
where $J=(J_1J_2\cdots J_r)$ is a concatenation of compositions $J_p$ of weight $i_p$ (by definition
of the refinement order).

Applying \cite[Prop. 4.9]{NCSF1}, we get the expression
\begin{equation}
K_I=\sum_{J\ge I}\frac{(-1)^{\ell(J)-\ell(I)}}{\ell(J,I)}S^J,
\end{equation}
where $\ell(J,I)=\prod_{p=1}^r\ell(J_p)$.

\section{Extension to $\WQSym^*$}

The identification of $\Sym_n$ with the (opposite) descent algebra of $\SG_n$ can be refined
as follows.

We have seen that $\WQSym^*$ can be identified with the (opposite) Solomon-Tits algebra. The dual basis
$\N_u=\M_u^*$ of the monomial basis of $\WQSym$ has the internal product rule \cite{NT06}
\begin{equation}
\N_u*\N_v = \N_{\pack{u\choose v}},
\end{equation}
and the embedding of the Solomon algebra into the Solomon-Tits algebra is given by
the Hopf embedding of $\Sym$ into $\WQSym^*$
\begin{equation}
S^I \mapsto \sum_{\ev(u)=I}\N_u.
\end{equation}
For example, $S^{21}=\N_{112}+\N_{121}+\N_{211}$.

In particular, $S_n(kA)$ and the Eulerian idempotents can be interpreted as elements of $\WQSym_n^*$, and
one can define a new basis
\begin{equation}
\K_u := \N_u * E_n^{[r]} \quad (r=\ell(\ev(u))=\max(u))
\end{equation}
which will be a simultaneous eigenbasis for the modified Adams operations
$\psi_k(F) := F*(\sigma_1)^k$.

The closed expressions given in $\Sym$ can be readily extended to $\WQSym^*$
thanks to the following lemma.

\begin{lemma}\label{lem:perm}
Define a right action of $\SG_n$ on $\WQSym_n^*$ by
\begin{equation}
\N_u\cdot\sigma := \N_{u\sigma},\ \text{where}\ u\sigma = u_{\sigma(1)}u_{\sigma(2)}\cdots u_{\sigma(n)}.
\end{equation}
Then, for any $I\vDash n$,
\begin{equation}
\N_{u\sigma}*S^I = (\N_u*S^I)\cdot \sigma.
\end{equation}
\end{lemma}
\Proof If $S^I$ contains $\N_v$, it contains the $\N_{v\tau}$ for all permutations $\tau$,
and
\begin{equation}\label{eq:permpack}
\pack{u\tau\choose v} = \pack{u\choose v\tau^{-1}}\cdot\tau.
\end{equation}
\qed

For example, with $u=111122$, $v=212211$, $\tau=451623$,
we have $u\tau = 121211$, $v\tau^{-1}=211212$,
$\pack{121211\choose 212211} =232411$, $\pack{111122\choose 211212}=211234$,
and $211234\tau = 232411$.

This implies\footnote{
Incidentally, this also proves the existence of the descent algebra. If one denotes by $\phi$
the dual of the inclusion map $\FQSym\rightarrow\WQSym$, which is given by
$\phi(\N_u)=\F_{\std(u)}$, then $\phi(S^I*\N_v) = \phi(S^I)\cdot\std(v)$, so that
$\phi(S^I*\N_v)=\phi(S^I)*\phi(\N_v)$.
}
that $(f\cdot\sigma)*g =(f*g)\cdot\sigma$ for all $f\in\WQSym^*_n,
g\in\Sym_n$ and $\sigma \in\SG_n$.

Hence,
\begin{equation}
\psi_k(\N_u) = \sum_{v\ge u}\beta_k(v,u) \N_v,
\end{equation}
where $\beta_k(v,u):=\beta_k(\ev(v),\ev(u))$, and
\begin{equation}
\K_u =\sum_{v\ge u}\frac{(-1)^{\ell(\ev(v))-\ell(\ev(u))}}{\ell(\ev(v),\ev(u))}\N_v,
\end{equation}
where the order on packed words $u,v$ is the {\it reverse} refinement order on the corresponding set compositions $\sigma,\pi$.

Given a spreadability system and a sequence $(X_i)$ of random variables, and defining the linear map
$\hat\varphi$ as above by $\hat\varphi(\N_u)=\varphi_u(X_1,\ldots,X_n)$, the cumulants are
given by
\begin{equation}
K_u(X_1,\ldots,X_n)=\hat\varphi(\K_u)
\end{equation}
and we recover the relations between moments
and cumulants of \cite[Th. 4.7 and 4.8]{HL}.

By construction,
\begin{equation}
\K_u*\S_n(NA) = N^{\ell(\ev(u))}\K_u
\end{equation}
which is \cite[Th. 4.14]{HL}.
Indeed, according to \cite[Definition 4.1]{HL}, $\varphi_\pi(N.X_1,\ldots,N.X_n)=\hat\varphi(\N_u*S_n(NA))$.

Adapting the argument given at the end of \cite{NCSF7}, one can prove the following extension of the ``splitting formula'' of \cite{NCSF1}:
\begin{proposition}
If $f_1,f_2,\ldots,f_r\in\WQSym^*$ and $g\in\Sym$, then
\begin{equation}
(f_1f_2\cdots f_r)*g = \mu_r[(f_1\otimes\cdots\otimes f_r) *_r\Delta^rg].
\end{equation}
\end{proposition}
The same argument as in  \eqref{eq:mult} proves then the following product rule for the $\K_u$:
\begin{equation}
\N_u\N_v = \sum_w c_{u,v}^w \N_w\ \Rightarrow \ \K_u\K_v =  \sum_w c_{u,v}^w \K_w.
\end{equation}
That is, $\N_u\mapsto \K_u$ is an algebra automorphism.

\section{Partial cumulants}
The defining formula  \eqref{eq:phipi=sumksig}
of the cumulants can be inverted using the M\"obius function
\eqref{eq:moebiusmu}, but neither this nor the formula of Cartier and Good
\eqref{eq:omegacom} are suitable
for the efficient calculation of cumulants of higher orders.
In the case of exchangeability systems recursive formulas are available
which are more adequate for this purpose; see \cite[Proposition~3.9]{Leh04}.
In the classical case, the recursion reads as follows:
\begin{equation}
K(X_1,X_2,\dots,X_n) =
\IE X_1X_2\dotsm X_n - \sum_{\substack{A\subsetneqq [n]\\ 1\in A}} K_{\abs{A}}(X_i:i\in A) \IE\prod_{j\in A^c}X_j
\end{equation}
and in the univariate case it specifies to the familiar formula
\cite{RS}
\begin{equation}
\kappa_n = m_n - \sum_{k=1}^{n-1}\binom{n-1}{k-1}\kappa_km_{n-k} .
\end{equation}

In the free case it specifies
to the free Schwinger-Dyson equation \cite{MingoNica:2013:schwinger}
and
from the point of view of combinatorial Hopf algebras this
has been recently considered under the name of ``splitting process''
\cite{EbrahimiFardPatras:2016:splitting}.

Turning to our general setting we note that already in the case of monotone
probability we lack a simple recursive formula;
however Hasebe and Saigo~\cite{HS11b,HS11a} found
a good replacement in terms of differential equations.
This was generalized to spreadability systems in terms of
partial cumulants introduced in \cite[Section 6]{HL}
which are the images by $\hat\varphi$ of some interesting elements of
$\WQSym^*$ which we shall study in this section.

We start with the analogous questions in $\Sym$, which are simpler and imply easily
the general results in $\WQSym^*$.

Let $T=(t_1,\ldots,t_r)$ be a sequence of binomial elements (scalars), so that the noncommutative
symmetric functions of $t_jA$ are defined by
\begin{equation}
\sigma_1(t_jA) := \sigma_1(A)^{t_j}
\end{equation}
and the analogs of the formal multivariate moments \cite[Eq. (6.2)]{HL} are
\begin{equation}
S^I(T;A) := S_{i_1}(t_1A)\cdots S_{i_r}(t_rA).
\end{equation}
The (generic) cumulants are thus
\begin{equation}
K_I = \left.\frac{\partial^r}{\partial t_1   \partial t_2\cdots \partial t_r}\right|_{T=(0,\ldots,0)}S^I(T;A)
\end{equation}
Imitating \cite[Def. 6.1]{HL}, we define the partial cumulants  as
\begin{equation}
K_{I;j}^{(t_1,\ldots,t_{j-1},1,t_{j+1},\ldots,t_r)} := \left.\frac{\partial}{\partial t_j}\right|_{t_j=0}S^I(T;A).
\end{equation}
Recall that $\sigma_1 = \exp(\phi)$, where $\phi=\sum_{n\ge 1}{\frac1n\Phi_n} = E^{[1]}$. Therefore,
\begin{equation}
\left.\frac{\partial}{\partial t}\right|_{t=0}\sigma_1(A)^t = \phi,
\end{equation}
so that
\begin{equation}
\left.\frac{\partial}{\partial t_j}\right|_{t_j=0}S^I(T;A)
 = S_{i_1}(t_1A)\cdots S_{i_{j-1}}(t_{j_1}A)\phi_{i_j} S_{i_{j+1}}(t_{j+1}A)\cdots S_{i_r}(t_rA),
\end{equation}
and inserting
\begin{equation}
\phi_n = \sum_{H\vDash n}\frac{(-1)^{\ell(H)-1}}{\ell(H)}S^H =:  \sum_{H\vDash =n}\tilde\mu(H,(n))S^H ,
\end{equation}
we obtain the analog of \cite[Prop. 6.2]{HL}
\begin{equation}
K_{I;j}^{(t_1,\ldots,t_{j-1},1,t_{j+1},\ldots,t_r)}
= \sum_{K\vDash i_j}S^{I'HI''}((T',1^{\ell(H)},T'');A)\tilde\mu(H,(n)),
\end{equation}
where $T'=(t_1,\ldots,t_{j-1})$, $T''=(t_{j+1},\ldots,t_r)$ and $1^p$ means $(1,\ldots,1)$ ($p$ times).

Now,
\begin{equation}
\frac{\partial}{\partial t_j}S^I(T;A)
\end{equation}
is the homogeneous part of degree $n$ in
\begin{align}
S_{i_1}(t_1A)\cdots S_{i_{j-1}}(t_{j-1}A)
\frac{\partial}{\partial t_j}e^{t_j\phi}
S_{i_{j+1}}(t_{j+1}A)\cdots S_{i_r}(t_rA)\\
=S_{i_1}(t_1A)\cdots S_{i_{j-1}}(t_{j-1}A)
\phi\sigma_1^{t_j}
S_{i_{j+1}}(t_{j+1}A)\cdots S_{i_r}(t_rA)\\
=
S_{i_1}(t_1A)\cdots S_{i_{j-1}}(t_{j-1}A)
\sigma_1^{t_j}\phi
S_{i_{j+1}}(t_{j+1}A)\cdots S_{i_r}(t_rA),
\end{align}
the last two expressions being respectively equal to
\begin{equation}
\sum_{a=1}^{i_j} K_{(i_1,\ldots,i_{j-1},a,i_j-a,i_{j+1},\ldots,i_r);j}^{(t_1,\ldots,t_{j-1},1,t_j,\ldots,t_r)}
\end{equation}
and
\begin{equation}
\sum_{a=1}^{i_j} K_{(i_1,\ldots,i_{j-1},i_j-a,a,i_{j+1},\ldots,i_r);j+1}^{(t_1,\ldots,t_{j-1},t_j,1,\ldots,t_r)}
\end{equation}

These relations are then extended to $\WQSym^*$ by defining $\N_u(T)$ in such a way that
\begin{equation}
S^I(T;A) = \sum_{\ev(u)=I}\N_u(T),
\end{equation}
which implies that (cf. \cite[Th. 4.5]{HL})
\begin{equation}
\N_u(T) = \sum_{v\ge u}\beta_T(v,u)\N_v.
\end{equation}

\begin{example}{\rm With $I=(2,2,1)$ we have
\begin{align}
S^{221}(T;A)&= \left(t_1S_2+{t_1\choose 2}S^{11}\right)\left(t_2S_2+{t_2\choose 2}S^{11}\right)t_3S_1\\
&= t_1t_2t_3S^{221}+t_1{t_2\choose 2}t_3S^{2111}+{t_1\choose 2}t_2t_3S^{1121}+{t_1\choose 2}{t_2\choose 2}t_3S^{11111},
\end{align}
so that
\begin{align}
K_{(221);2}^{(t_1,1,t_3)}& =  \left(t_1S_2+{t_1\choose 2}S^{11}\right)(S_2-\frac12 S^{11})t_3S_1\\
&= t_1t_3S^{221}-\frac12t_1t_3s^{2111}+{t_1\choose 2}t_3S^{1121} -\frac12{t_1\choose 2}t_3S^{11111}.
\end{align}
Similarly,
\begin{equation}
S^{2111}(T;A)= \left(t_1S_2+{t_1\choose 2}S^{11}\right)t_2t_3t_4S^{111}
\end{equation}
and
\begin{equation}
K_{(2111);3}^{(t_1,t_2,1,t_4)}= \left(t_1S_2+{t_1\choose 2}S^{11}\right)t_2t_4S^{111}.
\end{equation}
We have indeed
\begin{align}
K_{(221);2}^{(t_1,1,t_3)}+K_{(2111);3}^{(t_1,t_2,1,t_3)}
&= t_1t_3S^{221}+t_1t_3(t_2-\frac12)S^{2111}+{t_1\choose 2}t_3S^{1121}+{t_1\choose 2}(t_2-\frac12)t_3S^{11111}\nonumber\\
&=\frac{\partial}{\partial t_2}S^{221}(T;A).
\end{align}
In $\WQSym^*$, this would translate for the set composition $\pi=24|15|3\leftrightarrow u=21312$ as
\begin{equation}
\N_{21312}(T;A) =
 t_1t_2t_3\N_{21312}+t_1{t_2\choose 2}t_3\N_{21413}
+{t_1\choose 2}t_2t_3\N_{31423}+{t_1\choose 2}{t_2\choose 2}t_3\N_{31524},
\end{equation}
and
\begin{equation}
\N_{21413}(T;A) =
 t_1t_2t_3t_4\N_{21413}+{t_1\choose 2}t_2t_3t_4\N_{31524},
\end{equation}
and the partial cumulants $K_{24|15|3;15}^{(t_1,1,t_3)}$, $K_{24|1|5|3;5}^{(t_1,t_2,1,t_4)}$
would be respectively
\begin{equation}
K_{21312;2}^{(t_1,1,t_3)} = t_1t_3\N_{21312}-\frac12t_1t_3\N_{21413}+{t_1\choose 2}t_3\N_{31423}-\frac12{t_1\choose 2}t_3\N_{31524}
\end{equation}
\begin{equation}
K_{21413;3}^{(t_1,t_2,1,t_4)} = t_1t_2t_4\N_{21413}+{t_1\choose 2}t_2t_4\N_{31524}
\end{equation}

}
\end{example}

\section{Left and right products with the Eulerian idempotents}

We have seen that the cumulant basis is given by the internal products
$\N_u*E^{[r]}$ where $r=\ell(\ev(u))=\max(u)$.

The aim of this section is to compute the internal products
\begin{equation}
\N_u*E^{[k]}\quad\text{and}\quad  E^{[k]}*\N_u
\end{equation}
for arbitrary $u$ and $k$.

Let $u$ be a packed word. Recall that $v$ is said to refine $u$ if for all
 $i<j$, $v_i>v_j \Longleftrightarrow u_i\ge u_j$ and $v_i=v_j\Longrightarrow u_i=u_j$.
In this case, we write $v\ge u$ or  $v\in \raff(u)$.
This is the usual notion of refinement on set compositions: each block of $u$ is a
union of \emph{consecutive} blocks of $v$.

We shall say that $v$ is a \emph{weak refinement} of $u$, and write
$v\succeq u$ or $v\in \raffbis(u)$, if for all  $i<j$, $v_i=v_j\Longrightarrow u_i=u_j$.
On set compositions, this means that each block of $u$ is a union of blocks of $v$.

For example,
\begin{equation}
\begin{split}
& \raff(122) = \{122, 123, 132\}, \\
&\raffbis(122) = \{122,211,123,132,213,231,312,321\}.
\end{split}
\end{equation}

Thus, the word $w=\pack{u\choose v}$ is finer than $u$ and is such that
$w_i=w_j \Longrightarrow v_i=v_j$.

We shall compute the generating functions
\begin{align}
U_v(t) &= \sigma_1^t*\N_v = \sum_r t^r E^{[r]}*\N_v =  \sum_u {t\choose\max(u)}\N_u*\N_v\\
&=\sum_{w\in\raffbis(v)}\left( \sum_{u\in U(v,w)}{t\choose\max(u)}\right)\N_w,
\end{align}
where
\begin{equation}
U(v,w)=\left\{u|\pack{u\choose v}=w\right\}
\end{equation}
and
\begin{align}
V_u(t) &= \N_u*\sigma_1^t = \sum_r t^r \N_u*E^{[r]} =  \sum_v {t\choose\max(v)}\N_u*\N_v\\
&=\sum_{w\ge u}\left( \sum_{v\in V(u,w)}{t\choose\max(v)}\right)\N_w,
\end{align}
where
\begin{equation}
V(u,w)=\left\{v|\pack{u\choose v}=w\right\}.
\end{equation}

When convenient, we shall freely identify packed words with their corresponding set compositions
without further notice.

\subsection{A closed formula for $U_v(t)$}

The  pairs $(v,w)$ such that $U(v,w)$  is nonempty are those such that $w\in\raffbis(v)$.

\begin{proposition}
Define a set composition $u_0=u_0(v,w)$ as the one obtained by merging two consecutive blocks
$p',p''$ of $w$ if the blocks of $v$ containing $p'$ are strictly to the left of those containing $p''$,
and iterating the process until no further blocks can be merged.

Then, $U(v,w)$ is the interval
\begin{equation}
U(v,w) = [u_0,w]\quad \text{(compositions finer than $u_0$ and coarser than $w$).}
\end{equation}
\end{proposition}

For example, let us compute
$U(13211, 15342)$. The corresponding set compositions are
$145|3|2$ and $1|5|3|4|2$. To construct $u_0$, one can
merge the second and third blocks of $w$, and also the fourth and the fifth ones,
which yields $u_0=1|35|24$.
Therefore,
\begin{equation}
U(13211, 15342) = [13232,15342] = \{13232, 14232, 14342, 15342\}.
\end{equation}

To compute $U(13211, 24315)$, the corresponding set compositions are
$145|3|2$ et $4|1|3|2|5$, we obtain $u_0=4|123|5$, so that
\begin{equation}
U(13211, 24315) = [22213,24315] = \{22213, 23214, 23314, 24315\}.
\end{equation}

Finally, with $U(13211, 13214)$, the set compositions are
 $145|3|2$ et $14|3|2|5$,  $u_0=1234|5$,
and
\begin{equation}
U(13211, 13214) = [11112,13214] = \{11112, 12113, 12213, 13214\}.
\end{equation}

\Proof It follows from the definition of $\pack{u\choose v}$ that if two elements
$i,j$ are in the same block of $w$, then, they must also be in the same block in $u$ and in $v$,
and otherwise $i$ is in a block strictly left to the block of $j$ in $v$ if and only if
either $i$ is in a block left to the block of $j$ in $u$, or $i$ and $j$ are in the same
block of $u$, and the block of $i$ if left to the block of $j$ in $v$.

Thus, any $u\in U(v,w)$ must be obtained by merging some consecutive parts in $w$,
so that $u\le w$. Moreover, to get $w$ as $\pack{u\choose v}$, one can only merge
blocks satisfying the constraints mentioned in the definition of $u_0$, so that $u\ge u_0$.
\qed

So, $U(v,w)$ is the interval $[u_0,w]$. This interval is clearly a boolean lattice.
Moreover, if we set $a_0=\max(u_0)$ and $a=\max(w)$, the number of elements of
this lattice such that $\max(u)=k$ is ${a-a_0\choose a-k}$, so that
\begin{equation}
\sum_{u\in U(v,w)}
{t\choose\max(u)}
=
\sum_{k=a_0}^a {a-a_0\choose a-k} {t\choose k}
= {t+a-a_0\choose a},
\end{equation}
and finally
\begin{equation}\label{eq:U}
U_v(t) = \sum_w {t+\max(w)-a_0(v,w)\choose\max(w)}\N_w
\end{equation}
where $a_0(v,w) = \max(u_0(v,w))$.

In particular, the coefficient of $\N_w$ in $E^{[1]}*\N_v$ is the coefficient of
$t\N_w$ in $U_v(t)$, which is
\begin{equation}
(-1)^{a_0-1}\frac{(a-a_0)!(a_0-1)!}{a!}= (-1)^{a_0-1}{\rm B}(a-a_0+1,a_0)
\end{equation}
where ${\rm B}$ is the Beta function. This is formula (7.3) of \cite{HL} for the Weisner coefficients.

\subsection{A closed formula for $V_u(t)$}

Let us now describe $V(u,w)$.
Since the packing process commutes with the right action of the symmetric group (see \eqref{eq:permpack}),
we can apply to $u$
the smallest permutation $\sigma$
such that $u\sigma$ is nondecreasing (i.e., $\sigma=\std(u)^{-1}$), so that
$\pack{u\sigma\choose v\sigma}=w\sigma$. We can therefore assume that $u$ is nondecreasing.

\begin{proposition}
Let $w^{(i)}=\pack(w_{j_1}w_{j_2}\cdots w_{j_p})$, where $\{j_1,\ldots,j_p\}=\{j|u_j=i\}$.
Then,
\begin{equation}
\sum_{v\in V(u,w)}\M_v = \M_{w^{(1)}}\M_{w^{(2)}}\cdots \M_{w^{(\max(u))}}.
\end{equation}
\end{proposition}

\Proof 	Note first that no relation is imposed between the letters of $v$ corresponding
to different letters of $u$. The only order  constraints are among places where $u$ has identical letters,
and these are the same as in the corresponding letters of $w$. This is precisely the definition of the convolution
on packed words, describing the product of the $\M$ basis. \qed

Since the map $\M_u\mapsto {t\choose u}$ is a character of $\WQSym$,
\begin{equation}
\sum_{v\in V(u,w)} \binom{t}{\max(v)}
= \prod_{i=1}^{\max(u)} \binom{t}{\max(w^{(i)})}.
\end{equation}
We have therefore
\begin{equation}
V_u(t) = \sum_{w\ge u}\prod_{i=1}^{\max(u)} \binom{t}{\max(w^{(i)})}\N_w.
\end{equation}

For example, take $u=1122$ and $w=2133$.
We have $w^{(1)}=21$ et $w^{(2)}=\pack(33)=11$.
The product is
\begin{equation}
\M_{21} \M_{11} = \M_{2111} + \M_{2122} + \M_{2133} + \M_{3122} + \M_{3211},
\end{equation}
and the set of  $v$ is
\begin{equation}
V(1122,2133) = \{2111, 2122, 2133, 3122, 3211\}.
\end{equation}
We can then see that
\begin{equation}
2\binom{t}{2} + 3\binom{t}{3} = \frac{t^2(t-1)}{2} = \binom t2 \binom t1,
\end{equation}
as claimed.

\subsection{Mixed cumulants}

The mixed cumulants described in Section 7 of \cite{HL} correspond to the elements $\K_u*\N_v=\N_u*E^{[\max(u)]}*\N_v$
of $\WQSym^*$.

We shall compute the generating series
\begin{equation}
\N_u * \sigma_1^t * \N_v=\sum_r \N_u*E^{[r]}*\N_v.
\end{equation}
This amounts to computing
\begin{equation}
\label{all}
 \sum_{u'\ge u}
           \prod_{i=1}^{\max(u)} \binom{t}{\max(u'^{(i)})} \N_{u'}*\N_v.
\end{equation}

Let us compute the coefficient of $\N_w$ in~\eqref{all}.
First, the packed words having a nonzero coefficient in this expansion
are those finer than $u$, which are weakly finer than $v$.
Let $w$ be such a word. The coefficient of $\N_w$ is obtained by summing the coefficients of all $u'\ge u$ such that
$\pack{u'\choose v}=w$. The set of those $u'$ is therefore the set
of packed words which are finer than $u$ and than
$u_0(v,w)$, and that are coarser than $w$.

This set is an intersection of boolean lattices, hence also a boolean lattice.
Moreover, the subwords of $u$ consisting of identical letters cannot interfere
with each other, so that this lattice is in fact the product of the lattices
obtained by restricting $v$ and $w$ to positions where $u$ has identical letters.

Thus, the result is obtained by applying~\eqref{eq:U}
on each set of positions where $u$ has identical letters since these
pieces are independent and $\N_{1^k}=S_n$ is neutral for $*$. \qed

The result is therefore
\begin{equation}
\N_u * \sigma_1^t * \N_v
 = \sum_{w\in W(u,v)}
     \prod_{i=1}^{\max(u)} \binom{t+\max(w^{(i)})-a(v^{(i)},w^{(i)})}{\max(w^{(i)})} \N_{w},
\end{equation}
where $a$ is as previously defined, and
$W(u,v)$
is the set of packed words finer than $u$ which have equal letters only at
positions where $u$ has equal letters, and
$v^{(i)}$ et $w^{(i)}$
are the subwords of $v$ and $w$ corresponding to the positions of the letter $i$ in $u$.

\medskip
\begin{example}{\rm Let $u=11122$ and $v=12234$.

Take $w=23145$. We form the packed words of the restrictions of    $v$ and
$w$ to the positions where $u$ has equal letters, which gives
for the first block $v^{(1)}=122$ and $w^{(1)}=231$,
with a contribution of  ${t+1\choose 3}$  for this factor
(since $u_0^{(1)}=221$) and for the second,
$v^{(2)}=12$ et $w^{(2)}=12$, hence a contribution of ${t+1\choose 2}$ (since
$u_0^{(2)}=11$).

For $w=21354$, we have respective contributions of ${t+1\choose 3}$ and
${t\choose 2}$ since $u_0^{(1)}=212$ and $u_0^{(2)}=21$.

For $w=31245$,  we have respective contributions  of ${t\choose 3}$ and
${t+1\choose 2}$ since $u_0^{(1)}=312$ and $u_0^{(2)}=11$.

For  $w=31254$,  we have respective contributions  of ${t\choose 3}$ and
$\binom{t}{2}$ since $u_0^{(1)}=312$ and $u_0^{(2)}=21$.

Finally, for  $w=12243$,  we have respective contributions of ${t+1\choose 2}$ and
${t\choose 2}$ since $u_0^{(1)}=111$ and $u_0^{(2)}=21$.
}
\end{example}

\section{Miscellaneous remarks}

We have already noticed
\begin{equation}
\pack\gf {u\cdot \sigma}{v\cdot \sigma} = \pack\gf uv\cdot\sigma
\end{equation}
for any permutation $\sigma$.

Taking the smallest permutations sorting $u$ ($\sigma=\std(u)^{-1}$),
we can restrict to the case where $u$ is nondecreasing. With a second
permutation, we can restrict to the case where $v$ is nondecreasing
on positions where $u$ has equal letters.
Moreover, if
\begin{equation}
\pack\gf u{v_1} = \pack\gf u{v_2},
\end{equation}
then
\begin{equation}
\pack\gf {u'}{v_1} = \pack\gf {u'}{v_2}
\end{equation}
for all $u'$ finer than $u$. In the latter argument, it is thus possible
to restrict to the case where $u$ is a shifted concatenation of packed words,
one per block of equal letters of $u$.

\section{A generalization of Goldberg's formula}

The product of exponentials
\begin{equation}
g =  e^{a_1}e^{a_2}\cdots = \sum_{u\ {\rm packed}}\frac1{u!}\M_u(A), \quad (u! := \prod_i |u|_i!)
\end{equation}
is naturally an element of the completion $\widehat{\WQSym}$ of $\WQSym$. It is grouplike for the coproduct of $\WQSym$, which in this case
coincides with the coproduct of $\KK\llangle A\rrangle$.
The Hausdorff series
\begin{equation}
H(a_1,a_2,\ldots) = \log g = \sum_u c_u \M_u
\end{equation}
is thus also an element of $\widehat{\WQSym}$. The coefficient $c_u$ can be computed by Goldberg's formula, of which
new proofs respectively based on arguments from combinatorial Hopf algebras and of noncommutative probability
have been recently given in \cite{FPT} and \cite{HL}.

The homogeneous component $H_n$ of $H$ is the image of $g_n$ by the first Eulerian idempotent $e_n^{[1]}$, {\it aka} Solomon's
idempotent. More generally, the image of $g_n$ by $e_n^{[k]}$ is the coefficient of $t^k$ in $g^t$. Define polynomials $c_u(t)$ by
\begin{equation}
g^t =  (e^{a_1}e^{a_2}\cdots)^t =:\sum_u c_u(t)\M_u(A).
\end{equation}
Since $g$ is grouplike, we can, as in \cite{FPT}, define an injective morphism of Hopf algebras
\begin{align}
\varphi:\ & \Sym\longrightarrow \WQSym\\
&           S_n\longmapsto g_n
\end{align}
Then, $g^t =\varphi(\sigma_1^t)$ and
\begin{align}
c_u(t) &= \< \N_u,\varphi(\sigma_1^t)\> =\<\varphi^\dagger(\N_u),\sigma_1^t\>\\
&= \sum_I {t\choose \ell(I)}\<\varphi^\dagger(\N_u),S^I\>
\end{align}
where $\varphi^\dagger:\ \WQSym^*\rightarrow QSym$ is the adjoint map, and $\<\cdot,\cdot\>$
is the duality bracket.

Now, recall that the coproduct of $\N_u$ is given by \cite{NT06}
\begin{equation}
\Delta\N_u = \sum_{u=u_1\cdot u_2}\N_{\pack(u_1)}\otimes \N_{\pack(u_2)}
\end{equation}
(deconcatenation). We can omit the packing operation in this formula
if we make the convention that $\N_w=\N_u$ if $u=\pack(w)$. Then,
since $\varphi$, and hence also $\varphi^\dagger$ are morphisms of
Hopf algebras, for a composition $I=(i_1,\ldots,i_r)$,
\begin{equation}
\<\varphi^\dagger(\N_u),S^I\>=\prod_{k=1}^r\<\varphi^\dagger(\N_{u_k}),S^{i_k}\>
\end{equation}
where $u=u_1u_2\cdots u_r$ with $|u_k|=i_k$ for all $k$. Moreover, this is nonzero
only if all the $u_k$ are nondecreasing, in which case the result is
$1/(u_1!\cdots u_r!)$.

Let $u=u_1\cdots u_r$ be the minimal factorization of $u$ into nondecreasing words
({\it i.e.}, such that the last letter of each $u_i$ is strictly greater than the first one
of $u_{i+1}$), and let $I=(|u_1|,\ldots,|u_r|)$.

Let also $J=(j_1,\ldots,j_s)$ be the composition obtained by factoring $u$ into maximal blocks of
identical letters,
\begin{equation}
u = b_1^{j_1}b_2^{j_2}\cdots b_s^{j_s}.
\end{equation}
Then, it is worth noticing that $c_u(t)$ is the $(t,\E)$ specialization of
\begin{equation}
c_u(X;A) = \sum_{K\ge I} M_K(X)S^{K\vee J} = \sum_{H\ge J}\left(\sum_{K\vee J=H}M_K(X)\right)S^H(A) \quad\in QSym\otimes \Sym ,
\end{equation}
where $\vee$ denotes the join in the lattice of compositions of $n$.
For a binomial element $t$, $M_K(t)={t\choose\ell(K)}$, and the (virtual) exponential alphabet $\E$ is defined
by $S_k(\E)=\frac1{k!}$.
The set $\{K|K\vee J=H\}$ is a boolean lattice: it is the interval $[I\vee L,H]$ where  $\Des(L)=\Des(H)\backslash\Des(J)$.
Therefore,
\begin{equation}\label{eq:cu}
c_u(t) = \sum_{H\ge J}{t+\ell(J)-\ell(I)\choose\ell(H)}S^H(\E).
\end{equation}

For example, with $u=113223$, we have $I=33$, $J=2121$,
\begin{align}
c_u(X,A)=& (M_{33}+M_{213}+ M_{321}  + M_{2121})S^{2121}\nonumber\\
& + (M_{123} + M_{1113} + M_{1221} + M_{11121})S^{11121}\nonumber\\
& + (M_{312} + M_{3111} + M_{2112} + M_{21111})S^{21111}\nonumber\\
& +( M_{1212} + M_{12111} + M_{11112} + M_{111111})S^{111111}
\end{align}
whose  $(t,\E)$-specialization  reduces, by binomial convolution,  to
\begin{equation}
{t+2 \choose 4}\frac1{2!1!2!1!} +{t+2 \choose 5 }\frac1{1!1!1!2!1!} +{t+2 \choose 5}\frac1{2!1!1!1!1!} +{t+2 \choose 6 }\frac1{1!1!1!1!1!1!}
\end{equation}
This is the image of the polynomial
\begin{equation}
\left[\frac{t(2t^2+t)}{2}\right]^2 =: E^{2121}(t)
\end{equation}
under the linear substitution $t^k\mapsto {t+2\choose k}$.

Define a linear map ${\mathcal F}_s:\ t^k\mapsto {t+s\choose k}$, and associate to a composition $J$
the product of normalized Eulerian polynomials
\begin{equation}
E^J(t) := \prod_{k=1}^{\ell(J)}\frac{t E_{j_k}(t,t+1)}{j_k!}
\end{equation}
as in Goldberg's formula \cite[Theorem 3.11]{Reu},
where
\begin{equation}
E_n(x,y)=\sum_{\sigma\in\SG_n}x^{d(\sigma)}y^{n-d(\sigma)},
\end{equation}
and $d(\sigma)$ is the number of descents of $\sigma$.

\begin{theorem}
The $t$-Goldberg coefficient $c_u(t)$ is
\begin{equation}\label{eq:gg}
c_u(t) = {\mathcal F}_{s-r}(E^J(t)),
\end{equation}
where $r=\ell(I)$, $s=\ell(J)$, and $I,J$ are the compositions recording  respectively the lengths of the nondecreasing runs
and of the maximal blocks of identical letters of $u$.
\end{theorem}

The coefficient of $t$ in ${\mathcal F}_{s-r}(t^k)$ is
\begin{equation}
\begin{split}
(-1)^{k-s+r-1}\frac{(s-r)!(k-s+r-1)!}{k!}&=(-1))^{k-s+r-1}{\rm B}(s-r+1,k-s+r)\\
&=\int_{-1}^0t^k\cdot t^{r-1}(1+t)^{s-r}\frac{dt}{t^s}
\end{split}
\end{equation}
(since $k\ge s$ and $r\ge 1$), which applied to \eqref{eq:gg} yields the classical form of Goldberg's formula:
\begin{equation}
c_u = [t]c_u(t) = \int_{-1}^0 t^{r-1}(1+t)^{s-r} \prod_{k=1}^{s}\frac{ E_{j_k}(t,t+1)}{j_k!}dt.
\end{equation}
Note that $r-1$ is the number of descents of $u$, and $s-r$ its number of rises.

\section{Mixed cumulants and Goldberg coefficients}

It remains to explain the occurrence of Goldberg coefficients in the expansion of mixed
cumulants on the cumulant basis.

As before, we shall work with the $t$-analogues.

Let $u$ be a packed word.
Define as above two compositions $I(u)$ and $J(u)$ recording the lengths
of the maximal nondecreasing factors and of the maximal blocks of identical letters of $u$.
For example,
\begin{equation}
I(31121) = (1,3,1) \text{\quad and \quad} J(31121)=(1,2,1,1).
\end{equation}
Note that $J(u)\ge I(u)$.

Following \cite{HL}, with a pair of words such that $v\succeq u$, we associate their
\emph{refinement word} $m(u,v)$ defined as follows: $|m(u,v)|=\max(v)$, and
$m_i=u_j$ whenever $v_j=i$. For example,
\begin{equation}
m(12113,41223)= 2131,\
m(111123,353241) = 31121.
\end{equation}

We can now reformulate \eqref{eq:U} as
\begin{equation}\label{eq:newU}
U_v(t) = \sigma_1^t * \N_v
 = \sum_{ \ogf{w\in\raffbis(v)}{m=m(v,w)} } \binom{t+\ell(J(m))-\ell(I(m))}{|J(m)|} \N_w.
\end{equation}
For example,
\begin{equation}
\begin{split}
U_{112}(t) &=
\binom{t+1}{2} \N_{112} + \binom{t}{2} \N_{221} +
\binom{t+1}{3} \N_{123} + \binom{t+1}{3} \N_{132} \\
&+
\binom{t+1}{3} \N_{213} + \binom{t}{3} \N_{231} +
\binom{t+1}{3} \N_{312} + \binom{t}{3} \N_{321}.
\end{split}
\end{equation}
Computing $U_{1122}(t)$, one finds a sum of 38 terms, corresponding
to the packed words of which the first two or the last two letters can be identical.
There are 7 different coefficients in this expansion, given below together
with their associated words:
\begin{equation}
\begin{split}
\binom{t+2}{4} &: \{1324, 1342, 3124, 3142\} \\
\binom{t+1}{4} &: \{1234, 1243, 1423, 1432, 2134, 2143, 2314, 2341, 2413, \\
              &\ \ \ 2431, 3214, 3241, 4123, 4132, 4213, 4231\} \\
\binom{t+1}{3} &: [1123, 1132, 1233, 1322, 2133, 2213, 2231, 3122] \\
\binom{t+1}{2} &: \{1122\} \\
\binom{t}{4}   &: \{3412, 3421, 4312, 4321\} \\
\binom{t}{3}   &: \{2311, 3211, 3312, 3321\} \\
\binom{t}{2}   &: \{2211\}
\end{split}
\end{equation}

Let us now establish the equivalence of \eqref{eq:U} and \eqref{eq:newU}.
We have to prove that for all
 $v$ and $w$
such that $w\in\raffbis(v)$ :
\begin{equation}
\binom{t+\max(w)-a_0(v,w)}{\max(w)} =
\binom{t+\ell(J(m))-\ell(I(m))}{|J(m)|}.
\end{equation}
First of all, it is clear, by definition, that $|J(m)|=\max(w)$.
Moreover, $\max(w)-a_0(v,w)$ is the number of blocks of $w$ (regarded as a
set composition) whose all elements are all strictly to the left in $v$ of all
elements of the next block. This amounts precisely to merging two parts of $J(m)$
whenever the corresponding values in $m$ values are in increasing order.
Thus, $\max(w)-a_0(v,w)=\ell(J(m))-\ell(I(m))$.

Consider now the expansion of $U_v(t)$ on the basis $\K_w$.
Recall the transition matrices between $\N$ and $\K$.

\begin{equation}
\K_v = \sum_{w\in\raff(v)}
        \frac{(-1)^{\max(w)-\max(v)}}{\pev(m(u,v))} \N_w,
\end{equation}
\def\pev{{\pi_{\rm ev}}}
where $\pev(m)=\prod_i|m|_i$.

Notice that by definition
\begin{equation}
\K_v = [t^{\max(v)}] \N_v *\sigma_1^t.
\end{equation}
Indeed, to obtain the correct power of $t$, one must select in each
$\binom{t}{M}$ the coefficient of  $t$, which is $\frac{(-1)^{M-1}}{M}$.

In the other direction,
\begin{equation}
\N_v = \sum_{w\in\raff(v)}
        \frac{1}{\pev(m(u,v))!} \K_w,
\end{equation}
where $\pev(m)!=\prod_i|m|_i!$.

Ths yields for the expansion of $U_v(t)$ on the basis $\K$
\begin{equation}
U_v(t) = \sum_{x\in\raffbis(v)} c_{m(v,x)}(t) \K_x,
\end{equation}
where $c_{m(v,x)(t)}$ is defined in Eq.~\eqref{eq:cu}.

Indeed, to go from the expression of $U_v$ on the $\N_w$ to its
expression on the $\K_x$, notice that
the $\N_w$ contributing to a given $\K_x$ are those such that $w\le x$ and
$w\succeq v$.
This is a boolean lattice, an interval $[x_0,x]$ where $x_0$ is obtained by
merging consecutive blocks of $x$ whenever these blocks are contained in a
block of $v$.

We must therefore evaluate
\begin{equation}
\sum_{w\in [x_0,x]} \frac{1}{\pev(m(w,x))!}
\binom{t+\ell(J(m(v,w)))-\ell(I(m(v,w)))}{|J(m(v,w))|}.
\end{equation}
For a $w\in[x_0,x]$, it is easy to compute $m(v,w)$. The set of those
$m(v,w)$ is the set of words obtained from $m(v,x)$ by replacing blocks
$i^r$ of consecutive identical letters by smaller blocks $i^p$, $1\le p\le r$.
On this set,
$\ell(J(m))-\ell(I(m))$ is constant,
 $H=\ev(m(w,x))$
runs over the set of compositions finer than $\ev(m(x_0,x))$,
$1/\ev(m(w,x))!=S^H(\E)$ and $|J(m(v,w))|=\max(w)=\ell(H)$.

For example, let $v=111123$ and $x=356241$.
Then, $x_0=244231$.
The contribution to $\K_x$ in $U_v(t)$
come from the four words
\begin{equation}
\{ 244231, 245231, 355241, 356241\}.
\end{equation}
Their words $m(v,w)$ are respectively $3121$, $31211$, $31121$ and $311211$.
The  $m(w,x)$ are $122344$, $122345$, $123455$ and $123456$.
The sum of all contributions is
\begin{equation}\label{ex:gm}
\frac{1}{4}\binom{t+1}{4} + \frac{1}{2}\binom{t+1}{5} +
\frac{1}{2}\binom{t+1}{5} + \binom{t+1}{6}.
\end{equation}
This is the same as $c_u=c_{311211}(t)$.
Indeed,  $I(u)=(1,3,2)$ et $J(u)=(1,2,1,2)$, so that we get
\begin{equation}
\sum_{H\geq 1212} \binom{t+1}{\ell(H)} S^H(\E)
\end{equation}
which yields the same expression as \eqref{ex:gm}.

Finally, one can give the general formula for
$\N_u * \sigma_1^t * \N_v$ on the $\K_w$.
This is immediate, since the transition from $\N$ to $\K$
does not change the structure of the product. We get
\begin{equation}
\N_u * \sigma_1^t * \N_v
 = \sum_{w\in W(u,v)}
     \prod_{i=1}^{\max(u)} c_{m^{(i)}}(t) \K_w,
\end{equation}
where $c_u(t)$ are the  $t$-Goldberg coefficients, and
$m^{(i)}=m(v^{(i)},w^{(i)})$.

\footnotesize
\bibliographystyle{amsalpha}
\bibliography{hopfcumu}

\end{document}

config: arxiv


TITLE: Algorithms for mutually orthogonal latin squares - a correct one?
QUESTION [1 upvotes]: I am very interested in using mutually orthogonal latin squares (MOLS) to reduce the number of test cases but I struggle to find a way how to implement the algorithm. In an article published in a foreign language, I did find a formula including the source code.
This is said to be a formula to construct a series of squares (if $n$ is a prime number):
For $k, i, j \in \{1,2, \cdots, n\}$
$$A_k(i, j) = [k \cdot (i-1) + (j-1)] \mod n$$
Is this correct? I tried to find other sources but failed.
If this is correct, what is meant by $k$, $i$ and $j$? In the code, they pass "size" of the latin square to all of these variables.

REPLY [4 votes]: The info given in the OP is almost correct. As noted in Jyrki Lahtonen's answer, $k$ cannot be congruent to $n$.
In $A_k(i,j)$, $k$ is the index of the square: the algorithm generates $n-1$ MOLS, one for each $k$ in $\{1,\cdots,n-1\}$. Using normal matrix conventions, $i$ is the row index and $j$ is the column index, but the formula still works if those are swapped.
Here's some rudimentary Python code that uses that formula to generate a set of mutually orthogonal Latin squares (MOLS) and then tests each combination for orthogonality. This code was originally developed using Python 2.6 but it runs correctly on Python 3.
In this code, $i$ and $j$ run from 0 to $n-1$ and $k$ runs from 1 to $n-1$; note that $n$ must be prime.
If you pass a composite value for $n$, the program will generate some Latin squares, and if that $n$ is odd you'll get some pairs of MOLS, which is useful if you want to construct magic squares.

""" Generate sets of mutually orthogonal Latin squares (MOLS)

    See https://math.stackexchange.com/a/1624875/207316

    Written by PM 2Ring 2016.01.24
    Updated 2021.01.31
"""

#from __future__ import print_function

import sys

def show_grid(g):
    for row in g:
        print(row)
    print()

def test_mols(g):
    """ Check that all entries in g are unique """
    a = set()
    for row in g:
        a.update(row)
    return len(a) == len(g) ** 2

def mols(n):
    """ Generate a set of mutually orthogonal Latin squares
        n must be prime
    """
    r = range(n)

    #Generate each Latin square
    allgrids = []
    for k in range(1, n):
        grid = []
        for i in r:
            row = []
            for j in r:
                a = (k*i + j) % n
                row.append(a)
            grid.append(row)

        # Test that there are no repeated items in the 1st column.
        # This test is unnecessary for prime n, but it lets us
        # produce some pairs of MOLS for odd composite n
        if len(set([row[0] for row in grid])) == n:
            allgrids.append(grid)

    for i, g in enumerate(allgrids):
        print(i)
        show_grid(g)

    print('- ' * 20 + '\n')

    # Combine the squares to show their orthogonality
    m = len(allgrids)
    for i in range(m):
        g0 = allgrids[i]
        for j in range(i+1, m):
            g1 = allgrids[j]
            newgrid = []
            for r0, r1 in zip(g0, g1):
                newgrid.append(list(zip(r0, r1)))
            print(i, j, test_mols(newgrid))
            show_grid(newgrid)

def main():
    # Get order from command line, or use default
    n = int(sys.argv[1]) if len(sys.argv) > 1 else 5
    mols(n)

if __name__ == '__main__':
    main()

output for n=3

0
[0, 1, 2]
[1, 2, 0]
[2, 0, 1]

1
[0, 1, 2]
[2, 0, 1]
[1, 2, 0]

- - - - - - - - - - - - - - - - - - - -

0 1 True
[(0, 0), (1, 1), (2, 2)]
[(1, 2), (2, 0), (0, 1)]
[(2, 1), (0, 2), (1, 0)]

output for n=5
0
[0, 1, 2, 3, 4]
[1, 2, 3, 4, 0]
[2, 3, 4, 0, 1]
[3, 4, 0, 1, 2]
[4, 0, 1, 2, 3]

1
[0, 1, 2, 3, 4]
[2, 3, 4, 0, 1]
[4, 0, 1, 2, 3]
[1, 2, 3, 4, 0]
[3, 4, 0, 1, 2]

2
[0, 1, 2, 3, 4]
[3, 4, 0, 1, 2]
[1, 2, 3, 4, 0]
[4, 0, 1, 2, 3]
[2, 3, 4, 0, 1]

3
[0, 1, 2, 3, 4]
[4, 0, 1, 2, 3]
[3, 4, 0, 1, 2]
[2, 3, 4, 0, 1]
[1, 2, 3, 4, 0]

- - - - - - - - - - - - - - - - - - - -

0 1 True
[(0, 0), (1, 1), (2, 2), (3, 3), (4, 4)]
[(1, 2), (2, 3), (3, 4), (4, 0), (0, 1)]
[(2, 4), (3, 0), (4, 1), (0, 2), (1, 3)]
[(3, 1), (4, 2), (0, 3), (1, 4), (2, 0)]
[(4, 3), (0, 4), (1, 0), (2, 1), (3, 2)]

0 2 True
[(0, 0), (1, 1), (2, 2), (3, 3), (4, 4)]
[(1, 3), (2, 4), (3, 0), (4, 1), (0, 2)]
[(2, 1), (3, 2), (4, 3), (0, 4), (1, 0)]
[(3, 4), (4, 0), (0, 1), (1, 2), (2, 3)]
[(4, 2), (0, 3), (1, 4), (2, 0), (3, 1)]

0 3 True
[(0, 0), (1, 1), (2, 2), (3, 3), (4, 4)]
[(1, 4), (2, 0), (3, 1), (4, 2), (0, 3)]
[(2, 3), (3, 4), (4, 0), (0, 1), (1, 2)]
[(3, 2), (4, 3), (0, 4), (1, 0), (2, 1)]
[(4, 1), (0, 2), (1, 3), (2, 4), (3, 0)]

1 2 True
[(0, 0), (1, 1), (2, 2), (3, 3), (4, 4)]
[(2, 3), (3, 4), (4, 0), (0, 1), (1, 2)]
[(4, 1), (0, 2), (1, 3), (2, 4), (3, 0)]
[(1, 4), (2, 0), (3, 1), (4, 2), (0, 3)]
[(3, 2), (4, 3), (0, 4), (1, 0), (2, 1)]

1 3 True
[(0, 0), (1, 1), (2, 2), (3, 3), (4, 4)]
[(2, 4), (3, 0), (4, 1), (0, 2), (1, 3)]
[(4, 3), (0, 4), (1, 0), (2, 1), (3, 2)]
[(1, 2), (2, 3), (3, 4), (4, 0), (0, 1)]
[(3, 1), (4, 2), (0, 3), (1, 4), (2, 0)]

2 3 True
[(0, 0), (1, 1), (2, 2), (3, 3), (4, 4)]
[(3, 4), (4, 0), (0, 1), (1, 2), (2, 3)]
[(1, 3), (2, 4), (3, 0), (4, 1), (0, 2)]
[(4, 2), (0, 3), (1, 4), (2, 0), (3, 1)]
[(2, 1), (3, 2), (4, 3), (0, 4), (1, 0)]


TITLE: The author didn't write $\mathcal{T}$ is the minimum topology on $X$ which contains all basis elements. ("Topology 2nd Edition" by James R. Munkres)
QUESTION [0 upvotes]: I am reading "Topology 2nd Edition" by James R. Munkres.

Definition. If $X$ is a set, a basis for a topology on $X$ is a collection $\mathcal{B}$ of subsets $X$ (called basis elements) such that
(1) For each $x\in X$, there is at least one basis element $B$ containing $x$.
(2) If $x$ belongs to the intersection of two basis elements $B_1$ and $B_2$, then there is a basis element $B_3$ containing $x$ such that $B_3\subset B_1\cap B_2$.
If $\mathcal{B}$ satisfies these two conditions, then we define the topology $\mathcal{T}$ generated by $\mathcal{B}$ as follows: A subset $U$ of $X$ is said to be open in $X$ (that is, to be an element of $\mathcal{T}$) if for each $x\in U$, there is a basis element $B\in\mathcal{B}$ such that $x\in B$ and $B\subset U$. Note that each basis element is itself an element of $\mathcal{T}$.

If $\mathcal{T'}$ is a topology on $X$ such that $\mathcal{B}\subset\mathcal{T'}$, then $\mathcal{T}\subset\mathcal{T'}$.
So, $\mathcal{T}$ is the minimum topology on $X$ which contains all basis elements.
I think this fact is important, but the author didn't write that.
I wonder why the author didn't write that.
Any reason?
Is it unimportant to note this fact?
Because the author didn't write this fact, I thought $\mathcal{T}$ is a Topology but what $\mathcal{T}$ is.
I thought if $\mathcal{T}$ is special or not.

REPLY [1 votes]: I think I have a clue why the author didn't write it. It's just not the place to write it, because we can express it in more general manner: involving the notion of subbases. As you pointed out, the family of sets is a basis of some topology only if it satisfies two conditions. What if the family doesn't satisfy it? It isn't a basis of any topology (the formula you mentioned doesn't provide us with a topology), but we can consider the smallest (not only minimal!) topology containing all the sets from our family. This is a place for our new character: a subbasis.
If $\mathcal U$ is any family of subsets of $X$ then the family $\mathcal B$ of all intersections of finite subfamilies from $\mathcal U$ forms a basis (satisfies before mentioned two conditions) so it generates a topology.
Precisely, for any $\mathcal U\subset 2^X$ the family
$$\mathcal B = \mathscr B(\mathcal U) := \left\{
\bigcap U_0 \,|\,\mathcal U_0\subset U,\ \#\mathcal U_0<\infty \right\}\hspace{1cm}(*)$$ is a basis and the topology $\mathcal T$ generated by it is the smallest topology containing $\mathcal U$, i.e.
$$\mathcal T = \bigcap\{\tau\,|\, \tau\text{ is a topology on }X\text{ and }\mathcal U\subset \tau\}.$$ Then $\mathcal U$ is called a subbasis of $\mathcal T$.
In particular, if $\mathcal U$ is a basis, and $\mathcal B = \mathscr B(\mathcal U)$ is the basis generated by it (by the formula (*)) then both bases $\mathcal U$ and $\mathcal B$ generate the same topology, which implies that the topology generated by this basis is the smallest among all topologies possesing all sets from our basis.


TITLE: Can I increase pressure with sound waves?
QUESTION [2 upvotes]: I have a gas or a liquid in a container. Can I increase its pressure with sound waves or ultrasound or something? What I have in mind is that I can find the resonance frequency and make the molecules vibrate intensely. Is there a phenomenon like this? Any ideas of magnitude? Maybe a video demonstration? thanks!

REPLY [5 votes]: This is exactly what happens in a shockwave, so setting an explosion off would do the trick!  Doing it without an explosion is not very feasible though.
The hard part of trying to find the resonant frequency line that is that objects that are vibrating at their resonant frequency have a tendency to act like a speaker because they interact with the molecules of the air.  At the point where you start getting "interesting" results, it may be ear-splitting loud.  Sounds that we are used to hearing are tiny compared to the forces you'd be looking at.
A gas or a liquid would not develop a resonant frequency, but the chamber you put them in might have one, so we'll just start from the assumption that you have already found the resonant frequency to play with and ignore it for the rest of the though experiment.  It won't be important.  Let's pick a pressure which is meaningful: 970 mmHg.  This is the most pressure a man can exhale against a static load -- like blowing into a a balloon that's giving you too much resistance.  If I convert that pressure into the SI unit for pressure, the Pascal, we get 130,000Pa.  Using this calculator, I would need a sound volume of 196dB to attain that!
That sort of pressure is unattainable using sound waves in air.  A .30-06 rifle going off right next to your ear is 171dB, 3,00 times weaker than the sound you'd need for this experiment (remember that dB is a logarithmic scale).  You actually hit a limit around 194dB.  At 194dB, you can no longer form an undistorted wave because the trough of the wave actually creates a perfect vacuum.  So to create a modest pressure, one you could create using your own lungs and diaphragm, you would actually need a louder sound than the air supports!
But it does work.  The trick is that blast waves don't try to create a nice sinusoidal sound wave.  They merely create a large pressure gradient, pushing the air into a range where it is no longer effective to treat it as a linear transmission medium.  In fact, modern nuclear devices use this when they go off.  They use a large number of carefully shaped conventional explosives around the outside of the weapon to generate a shockwave which actually compresses the solid nuclear fuel in the middle.  They do some really clever tricks to focus this shockwave such that at the center you get enough compression to increase the density of the fuel past critical density and it starts the chain reaction.

REPLY [4 votes]: An ideal sound wave will alternately increase, then decrease, the pressure locally - with no mean shift in pressure. However, there is a small amount of dissipation of sound energy - meaning that the gas will slowly heat if subjected to sound. So if your container has a fixed volume, it is possible to (slightly) heat the gas inside with sound; theoretically this would increase the pressure.
It is not a very practical mechanism except in very specific circumstances.


TITLE: how to extend a set to an R-basis of V4(R)
QUESTION [0 upvotes]: I'm going through some questions in my book and there's one in particular that I can't really find explanations on. Any help would be really appreciated (with worked solutions if possible as I don't understand the topic of vector spaces completely). Thanks in advance!
The question:
Extend $\{(1,2,4,0),(1,0,0,5)\}$ to an $R$-basis of $V_4(R)$.

REPLY [0 votes]: First we observe the following:

$\textbf{Theorem:}$ Let $V$ be a vector space
  that is generated by a set $G$ containing exactly $n$ vectors, and let $L$ be a
  linearly independent subset of $V$ containing exactly $m$ vectors. Then $m \leq n$ and there exists a subset H of G containing exactly $n — m$ vectors such that $L \cup H $ generates V.
$\textbf{Corollary:}$  Let $V$ be a vector space with dimension n.
(a) Any finite generating set for V contains at least n vectors, and a generating
  set for V that contains exactly n vectors is a basis for V.
(b) Any linearly independent subset of V that contains exactly n vectors is
  a basis for V.
(c) $\textbf{Every linearly independent subset of V can be extended to a basis for
V}$

So first check whether your two vectors is linearly independent or not.
Fix any basis S for $\mathbb{R}^4$.
One method is adding two vectors from S  to your list and check if the 4 vectors are linearly independent, if not, then choose another two vectors until you have the 4 linearly independent vectors.
The second thing is fix any basis for $\mathbb{R}^4$ and combine these four to your two vectors. Now we eliminate the dependent vectors.


TITLE: Function problem involving limits and periodicity
QUESTION [0 upvotes]: Let $f: \Bbb R \to \Bbb R$ a continuous, non-constant and periodic function, such that $T \in \Bbb R-\Bbb Q$ is one of its periods. Show that the sequence $f(n)$ cannot be convergent, where $n \in \Bbb N$

REPLY [2 votes]: Let $m=\min f$, and $M=\max f$ (which exist as we can simpy consider the compact interval $[0,T]$). By assumption, $M>m$, and we find $x_m,x_M$ and $\epsilon>0$ such that $f(x)>\frac{2M+m}3$ for $|x-x_M|<\epsilon$ and $f(x)<\frac{M+2m}3$ for $|x-x_m|<\epsilon$.
As $\frac 1T$ is irrational, the set of fractional parts  $\{\frac nT\}$ is dense in $[0,1]$. Therefore, there are infinitely many $n$ such that $\{\frac nT\}-\{\frac {x_m}T\}<\frac \epsilon {|T|}$ and hence $f(n)<\frac{M+2m}3$. Likewise, there are infinitely many $n$ such that $f(n)>\frac{2M+m}3$. This is not compatible with $f(n)$ converging.


TITLE: How can I simplify this boolean expression?
QUESTION [2 upvotes]: Original expression: $$D=A’BCD+AB’C’D+AB’CD’+AB’CD+ABC’D’+ABC’D+ABCD’+ABCD$$
This is the simplest form I could find: $$B\cdot C\cdot D+A\cdot(B'\cdot D+B \cdot C'+CD')$$
It is equivalent to the solution. This is the solution I got from an online calculator: $$A \cdot B + A \cdot C + A \cdot D + B \cdot C \cdot D$$
How can I do this? Do I need to take a step back and then simplify more?

REPLY [1 votes]: $$A’BCD+AB’C’D+AB’CD’+\color{red}{AB’CD}+ABC’D’+\color{green}{ABC’D}+\color{blue}{ABCD’}+\color{orange}{ABCD} = \text{(Idempotence x 6)}$$
$$A’BCD+AB’C’D+AB’CD’+\color{red}{AB’CD+AB’CD}+ABC’D’+\color{green}{ABC’D+ABC’D}+\color{blue}{ABCD’+ABCD’}+\color{orange}{ABCD +ABCD+ABCD+ABCD}= \text{(Commutation)}$$
$$\color{red}{ABC’D’+ABC’D+ABCD’+ABCD}+\color{green}{AB’CD’+AB’CD+ABCD’+ABCD} +\color{blue}{AB’C’D +ABC’D+AB’CD+ABCD}+\color{orange}{A’BCD+ABCD}= \text{(Adjacency)}$$
$$\color{red}{ABC’+ABC}+\color{green}{AB’C+ABC} +\color{blue}{AB’D+ABD}+\color{orange}{BCD}= \text{(Adjacency)}$$
$$\color{red}{AB}+\color{green}{AC} +\color{blue}{AD}+\color{orange}{BCD}$$
(note: the colors help connect lines 1 and 2, and also lines 3,4,5, but the colors do not reveal how line 2 goes to line 3: that step is really just a matter of reordering all the terms by Commutation)


TITLE: Convergence of a sequence defined by $ c_{n+1}=\frac{r_{n+1}\sqrt{c_n}}{3}$
QUESTION [4 upvotes]: Define a sequence $(r_n)$ by $r_0=1$ and $r_{n+1}=(2/3)r_n+1$ for $n\geq 0$. Let the sequence $(c_n)$ be defined by $c_0=1/4$, and
  $$
c_{n+1}=\frac{r_{n+1}\sqrt{c_n}}{3}
$$
  for $n\geq 0$. Prove that $\lim_{n\to\infty}c_n$ exists.

It is not difficult to prove that the sequence $(r_n)$ converges to $3$ since it is bounded above and non-decreasing. I don't have a more useful estimate for $(c_n)$ than $c_{n+1}\leq \sqrt{c_n}$. How shall I go on?

REPLY [2 votes]: Enlightened by answers and comments, I have a proof.
One can see that $r_{n+1}-3=\frac{2}{3}(r_n-3),$ which implies by induction that
$$
r_{n}-3=(r_0-3)q^n,\quad q=2/3,
$$
and thus
$$
c_{n+1}=\left(1-\left(\frac{2}{3}\right)^{n+2}\right)\cdot\sqrt{c_n}=:b_n\sqrt{c_n}.
$$
On the other hand, $c_1\geq c_0$ by direct calculation. Suppose $c_{n+1}\geq c_n$. Then
$$
c_{n+2}=b_{n+1}\sqrt{c_{n+1}}\geq b_n\sqrt{c_{n+1}}\geq b_n\sqrt{c_n}=c_{n+1}.
$$
It follows that $(c_n)$ is a non-decreasing sequence.
All we need to do now is showing that $(c_n)$ is bounded above. We use induction again. Obviously $c_0\leq 1$. Suppose $c_n\leq 1$. Then
$$
c_{n+1}\leq\sqrt{c_n}\leq 1.
$$
We are done.