text:
\begin{document}
\title[Combinatorics of antiprism triangulations]
{Combinatorics of antiprism triangulations}

\author{Christos~A.~Athanasiadis}
\address{Department of Mathematics\\
National and Kapodistrian University of Athens\\
Panepistimioupolis\\
15784 Athens, Greece}
\email{caath@math.uoa.gr}

\author{Jan-Marten~Brunink}
\address{Universit\"at Osnabr\"uck\\
Fakult\"at f\"ur Mathematik\\
Albrechtstrasse 28A\\
49076 Osnabr\"uck, Germany}
\email{janmarten.brunink@uni-osnabrueck.de}

\author{Martina~Juhnke-Kubitzke}
\address{Universit\"at Osnabr\"uck\\
Fakult\"at f\"ur Mathematik\\
Albrechtstrasse 28A\\
49076 Osnabr\"uck, Germany}
\email{juhnke-kubitzke@uni-osnabrueck.de}

\date{June 18, 2020}
\thanks{ \textit{Key words and phrases}.
Simplicial complex, triangulation, antiprism,
face enumeration, $h$-polynomial, real-rootedness,
Lefschetz property.}

\begin{abstract}
The antiprism triangulation provides a natural way
to subdivide a simplicial complex $\Delta$, similar to
barycentric subdivision, which appeared independently
in combinatorial algebraic topology and computer
science. It can be defined as the simplicial complex
of chains of multi-pointed faces of $\Delta$, from
a combinatorial point of view, and by successively
applying the antiprism construction, or balanced
stellar subdivisions, on the faces of $\Delta$, from
a geometric point of view.

This paper studies enumerative invariants associated
to this triangulation, such as the transformation of
the $h$-vector of $\Delta$ under antiprism
triangulation, and algebraic properties of its
Stanley--Reisner ring. Among other results, it is
shown that the $h$-polynomial of the antiprism
triangulation of a simplex is real-rooted and that
the antiprism triangulation of $\Delta$ has the
almost strong Lefschetz property over $\RR$ for every
shellable complex $\Delta$. Several related open
problems are discussed.
\end{abstract}

\maketitle

\section{Introduction}
\label{sec:intro}

Barycentric subdivision provides a natural way to
triangulate a simplicial complex $\Delta$, of
fundamental importance in algebraic topology. Because
of its especially nice enumerative and algebraic
properties, it has also been studied intensely from
the point of view of enumerative and algebraic
combinatorics; see \cite{BW08, CKW18, KMS19, KN09,
KW08, NPT11}, \cite[Chapter~9]{Pet15}. For instance,
Brenti and Welker~\cite{BW08} described in explicit
combinatorial terms the transformation of the
$h$-vector (a fundamental enumerative invariant) of
$\Delta$, under barycentric subdivision, and showed
that the $h$-polynomial (the generating polynomial
for the $h$-vector) of the barycentric subdivision
of $\Delta$ has only real roots (and in particular,
log-concave and unimodal coefficients) for every
simplicial complex $\Delta$ with nonnegative
$h$-vector.

A similar, but combinatorially more intricate and
much less studied than barycentric subdivision,
way to subdivide $\Delta$ is provided by the
\emph{antiprism triangulation}, denoted here by
$\sd_\aA(\Delta)$. To give the reader a hint on
the comparison between the two triangulations, we
recall that the barycentric subdivision of a
geometric simplex $\Sigma$ can be constructed
by inserting a vertex in the interior of $\Sigma$
and coning over its proper faces, which have been
barycentrically subdivided by induction. The
antiprism triangulation $\sd_\aA(\Sigma)$ instead
can be constructed by inserting another simplex of
the same dimension in the interior of $\Sigma$,
whose vertices are in a given one-to-one
correspondence with those of $\Sigma$, and
joining each nonempty face of that simplex with
the antiprism triangulation of the complementary
face of $\Sigma$. Figure~\ref{fig:2simplex} shows
the antiprism triangulation of a 2-dimensional
simplex (the labeling of faces is explained in
Section~\ref{sec:atriang}). As an abstract
simplicial complex, the barycentric subdivision
of $\Delta$, denoted here by $\sd(\Delta)$,
has faces which correspond bijectively to the
ordered partitions of the faces of $\Delta$; in
particular, the vertices and facets of $\sd(\Delta)$
correspond bijectively to the nonempty faces and the
permutations of the facets of $\Delta$, respectively.
The faces of $\sd_\aA(\Delta)$ instead correspond
bijectively to certain multi-pointed ordered
partitions of the faces of $\Delta$; in particular,
the vertices and facets of $\sd_\aA(\Delta)$
correspond bijectively to the pointed faces and the
ordered partitions of the facets of $\Delta$,
respectively.

The antiprism triangulation was introduced
by Izmestiev and Joswig \cite{IJ03} as a technical
device in their effort to understand combinatorially
branched coverings of manifolds, and arose
independently and was studied under the name
\emph{chromatic subdivision} in computer science
(specifically, in theoretical distributed computing);
see \cite{Ko12} and references therein.
This paper aims to show that, as is the case with
barycentric subdivision, the antiprism
triangulation has very interesting enumerative and
algebraic properties and that its study leads to
combinatorial problems which are often more
challenging than the corresponding ones for the
barycentric subdivision. We denote by
$h(\Delta, x)$ the $h$-polynomial of a simplicial
complex $\Delta$ and by $\sigma_n$ the (abstract)
simplex on an $n$-element vertex set. Our main
motivation comes from the following conjectural
analogue of the main result of~\cite{BW08}.

\begin{figure}
\centering
  \includegraphics[width=0.7\textwidth]{2simplexcolor}
  \caption{Antiprism triangulation of the 2-simplex}
  \label{fig:2simplex}
\end{figure}

\begin{conjecture} \label{conj:main}
The polynomial $h(\sd_\aA(\Delta), x)$ is real-rooted
for every simplicial complex $\Delta$ with nonnegative
$h$-vector.
\end{conjecture}

This conjecture is part of the general problem to
understand when the $h$-polynomial of a triangulation
of a simplicial complex is real-rooted. The present
study of antiprism triangulations has partly motivated
the study of this problem for the much more general
class of uniform triangulations~\cite{Ath20+}. Although
we are unable to fully settle \Cref{conj:main} in this
paper, we reduce it to an interlacing relation
between the members of two concrete infinite sequences
of polynomials (see Conjecture~\ref{conj:thetaA}),
given the following important special case of the
conjecture and \cite[Theorem~1.2]{Ath20+}.

\begin{theorem} \label{thm:mainA}
The polynomial $h(\sd_\aA(\sigma_n), x)$ is real-rooted
and has a nonnegative, real-rooted and interlacing
symmetric decomposition with respect to $n-1$ for
every positive integer $n$.
\end{theorem}

We also prove the unimodality of $h(\sd_\aA(\Delta), x)$
for every Cohen--Macaulay simplicial complex $\Delta$
and show that the peak appears in the middle, by
studying Lefschetz properties of the Stanley--Reisner
ring of $\sd_\aA(\Delta)$. The following result is
an analogue of the main result of~\cite{KN09} for the
barycentric subdivision.

\begin{theorem} \label{thm:LefschetzUnimodal}
The complex $\sd_\aA(\Delta)$ has the almost strong
Lefschetz property over $\RR$ for every shellable
simplicial complex $\Delta$.

Moreover, for every $(n-1)$-dimensional
Cohen--Macaulay simplicial complex $\Delta$, the
$h$-vector of $\sd_\aA(\Delta)$ is unimodal, with
the peak being at position $n/2$, if $n$ is even,
and at $(n-1)/2$ or $(n+1)/2$, if $n$ is odd.
\end{theorem}

This paper is structured as follows. The antiprism
triangulation $\sd_\aA(\Delta)$ is described
combinatorially as an abstract simplicial complex
and defined geometrically as a triangulation, using
either the antiprism construction, or balanced
stellar subdivisions (crossing operations), in
Section~\ref{sec:atriang}. The antiprism construction
is defined in Section~\ref{sec:aconstr}, where its
face enumeration is studied within the framework of
uniform triangulations, introduced in~\cite{Ath20+}.
These results are then applied in
Section~\ref{sec:enumerate} to find combinatorial
interpretations and recurrences for the basic
enumerative invariants of the antiprism triangulation
of the simplex. The face enumeration of antiprism
triangulations turns out to be related to traditional
combinatorial themes, such as ordered set partitions,
colorings and the enumeration of permutations by
excedances (for example, the number of facets of
$\sd_\aA(\sigma_n)$ is equal to the number of ordered
partitions of an $n$-element set).
Section~\ref{sec:enumerate} also proves
Theorem~\ref{thm:mainA} and describes combinatorially
the transformation of the $h$-vector of a simplicial
complex, under antiprism triangulation. The proof of
Theorem~\ref{thm:mainA} is different from all proofs
of the corresponding result for the barycentric subdivision
known to the authors; it exploits the recurrence
for the $h$-polynomial of $\sd_\aA(\sigma_n)$ and
uses the concept of interlacing sequence of
polynomials. Theorem~\ref{thm:LefschetzUnimodal} is
proved in Section~\ref{sec:aLefschetz}; the method
generally follows those of \cite{KN09, Mur10}, with
certain complications and shortcuts.

Basic background and definitions, together with
some preliminary technical results, are included
in Section~\ref{sec:pre} for simplicial complexes,
their triangulations and face enumeration, and for
the unimodality and real-rootedness of polynomials
and their symmetric decompositions, and in
Section~\ref{sec:Lefschetz}
for Lefschetz properties of simplicial complexes.
Open problems, other than those proposed earlier
in the paper, and further directions for research
are discussed in Section~\ref{sec:direct}.

\section{Preliminaries}
\label{sec:pre}

This section includes preliminaries on simplicial
complexes and triangulations, their basic enumerative
invariants and the unimodality of polynomials and
related properties. Throughout this paper we set
$\NN := \{0, 1, 2,\dots\}$ and $[n] :=
\{1, 2,\dots,n\}$ for $n \in \NN$. We also denote
by $\fS_n$ the symmetric group of permutations of
$[n]$ and by $|V|$ and $2^V$ the cardinality and
the powerset, respectively, of a finite set $V$.

\subsection{Simplicial complexes}
\label{sec:complexes}

We start with several definitions and refer to Stanley's
book \cite{StaCCA} for background and more information.

Let $V$ be a finite set. An (abstract) \emph{simplicial
complex} $\Delta$ on the vertex set $V$
is a collection of subsets of $V$ that is closed under
inclusion and such that $\{v\} \in \Delta$ for every
$v\in V$. Throughout this article, we assume that all
simplicial complexes are finite. The elements of
$\Delta$ are called \emph{faces} and the inclusionwise
maximal ones are called \emph{facets}. The
\emph{dimension} of a face $F \in \Delta$ is defined
as $\dim(F) = |F|-1$; the \emph{dimension} of $\Delta$,
denoted by $\dim(\Delta)$, is the maximum dimension of
its faces. Zero-dimensional and one-dimensional faces
of $\Delta$ are called \emph{vertices} and \emph{edges},
respectively. We say that $\Delta$ is \emph{pure} if
all facets of $\Delta$ have the same dimension. As
in~\cite{Ath20+}, we denote by $\sigma_n$ the abstract
$(n-1)$-dimensional simplex $2^V$ on an $n$-element
vertex set $V$ (often taken to be $[n]$).

The \emph{cone} over $\Delta$ is the simplicial
complex consisting of the faces of $\Delta$, together
with all sets $F \cup \{u\}$ for $F \in \Delta$, where
$u \notin V$ is a new vertex, called the \emph{apex}.
We will denote this cone by $u \ast \Delta$. More
generally, the (simplicial) \emph{join} of two
simplicial complexes $\Delta_1$ and $\Delta_2$ with
disjoint vertex sets is defined as $\Delta_1 \ast
\Delta_2 = \{ F_1 \cup F_2~ :~ F_1 \in \Delta_1, F_2
\in \Delta_2\}$. Given a face $F \in \Delta$, the
\emph{link} and the \emph{star} of $F$ in $\Delta$
are defined as the simplicial complexes

\begin{center}
\begin{tabular}{l}
$\link_\Delta(F) \ = \ \{ G \in \Delta~:~ F \cup
G \in \Delta,\ F \cap G = \varnothing \}$ and \\
$\Star_\Delta(F) \ = \ \{ G \in \Delta~: ~F \cup
G \in \Delta\}$,
\end{tabular}
\end{center}

respectively. For $G_1, G_2,\dots,G_m \subseteq V$
we set

\begin{equation*}
\langle G_1, G_2,\dots,G_m\rangle \ = \
\{F~:~ F \subseteq G_i \text{ for some } i \in [m]
       \rangle.
\end{equation*}

In the sequel, $\Delta$ is a pure $(n-1)$-dimensional
simplicial complex with vertex set $V$ and $\FF$ is a
field. Let $A$ be the polynomial ring $\FF[x_v: v
\in V]$ and write $x_F = \prod_{v \in F} x_v$ for $F
\subseteq V$.
The \emph{Stanley-Reisner ring} (or \emph{face ring})
of $\Delta$ (over $\FF$) is defined as the quotient
ring $\FF[\Delta] = A/I_\Delta$, where $I_\Delta=
(x_F: F \subseteq V,\ F \not \in \Delta)$ is the
ideal of $A$ known as the \emph{Stanley-Reisner ideal}
(or \emph{face ideal}) of $\Delta$. The ring
$\FF[\Delta]$ is graded by degree; subscripts on
$\FF[\Delta]$ and its (standard) graded quotients
will always refer to homogeneous components.

A \emph{linear system of parameters} (l.s.o.p. for
short) for $\FF[\Delta]$ is a sequence
$\Theta=\theta_1,\dots,\theta_n$ of linear forms in
$\FF[\Delta]$ such that the quotient $\FF[\Delta] /
\Theta\FF[\Delta]$ has finite dimension, as a vector
space over $\FF$. The complex $\Delta$ is called
\emph{Cohen--Macaulay} over $\FF$ if $\FF[\Delta]$
is a free module over the polynomial ring
$\FF[\Theta]$ for some (equivalently, for every)
l.s.o.p. $\Theta$ for $\FF[\Delta]$ and
\emph{shellable} if there exists a linear ordering
$G_1, G_2,\dots,G_m$ of the facets of $\Delta$ such
that for each $2 \le j \le m$, the set

\begin{equation*}
\{F \subseteq G_j~:~ F \not\subseteq G_i \text{ for }
  1 \le i < j\}
\end{equation*}

has a unique minimal element, with respect to inclusion.
Even though shellable simplicial complexes constitute a
proper subclass of that of Cohen--Macaulay complexes,
the sets of possible $f$-vectors for the two classes
of simplicial complexes coincide (see, e.g.,
\cite[Theorem~3.3]{StaCCA}).

Given an $(n-1)$-dimensional simplicial complex
$\Delta$, the \emph{$f$-vector} of $\Delta$ is defined
as the sequence $f(\Delta) = (f_{-1}(\Delta),
f_0(\Delta),\dots,f_{n-1}(\Delta))$, where $f_i
(\Delta)$ denotes the number of $i$-dimensional faces
of $\Delta$. The \emph{$h$-vector} of $\Delta$ is
defined as $h(\Delta)=(h_0(\Delta),
h_1(\Delta),\dots,h_n(\Delta))$, where $h_i(\Delta)$
is given by the formula

\begin{equation*}
h_i(\Delta) \ = \ \sum_{j=0}^i(-1)^{i-j}{n-j \choose
             i-j} f_{j-1}(\Delta),
\end{equation*}

and $h(\Delta,x) = \sum_{i=0}^n h_i(\Delta)x^i$ is the
\emph{$h$-polynomial} of $\Delta$. Equivalently, the
latter can be defined by the formula

\begin{equation}
\label{eq:hdef}
h(\Delta, x) \ = \ \sum_{i=0}^n f_{i-1} (\Delta) \,
x^i (1-x)^{n-i} \ = \ \sum_{F \in \Delta} x^{|F|}
(1-x)^{n-|F|}.
\end{equation}

Assume now that $\Delta$ triangulates an
$(n-1)$-dimensional ball, meaning that the geometric
realization of $\Delta$ is homeomorphic to an
$(n-1)$-dimensional ball (we also say that $\Delta$
is a an $(n-1)$-dimensional simplicial ball). The
\emph{boundary} of $\Delta$ is then defined as

\begin{equation*}
\partial\Delta \ = \ \langle F \in \Delta~:~F\subseteq
G \text{ for a unique facet } G \in \Delta\rangle.
\end{equation*}

The set $\Delta^\circ = \Delta \sm \partial\Delta$
consists of the \emph{interior} faces of $\Delta$
and $h^\circ(\Delta,x)$ is defined by the
sum on the far right of (\ref{eq:hdef}) in which
$\Delta$ has been replaced by $\Delta^\circ$. The
following well known statement is a special case of
\cite[Lemma~6.2]{Sta87}.

\begin{proposition} \label{prop:h-ballcirc}
{\rm (\cite{Sta87})} We have $x^n h (\Delta,
1/x) = h^\circ(\Delta, x)$ for every
triangulation $\Delta$ of an $(n-1)$-dimensional
ball.
\end{proposition}

\subsection{Triangulations}
\label{sec:triang}

Let $\Delta$ and $\Delta'$ be simplicial complexes.
We say that $\Delta'$ is a \emph{triangulation} of
$\Delta$ if there exist geometric realizations $K'$
and $K$ of $\Delta'$ and $\Delta$, respectively, such
that $K'$ geometrically subdivides $K$. Let $L \in K$
be a simplex and $F$ be the corresponding face of
$\Delta$. Then, $K'$ restricts to a triangulation
$K'_L$ of $L$. The subcomplex $\Delta'_F$ of $\Delta'$
which corresponds to $K'_L$ is a triangulation of the
abstract simplex $2^F$, called the \emph{restriction}
of $\Delta'$ to $F$. The \emph{carrier} of a face
$G \in \Delta'$ is the smallest face $F \in \Delta$
such that $G \in \Delta'_F$.

A fundamental enumerative invariant of a
triangulation of a simplex is the \emph{local
$h$-polynomial}. Given a triangulation $\Gamma$ of an
$(n-1)$-dimensional simplex $2^V$, this polynomial is
defined \cite[Definition~2.1]{Sta92} by the formula

\begin{equation} \label{eq:deflocalh}
  \ell_V (\Gamma, x) \ = \sum_{F \subseteq V}
  \, (-1)^{n - |F|} \, h (\Gamma_F, x).
\end{equation}

By the principle of inclusion-exclusion,

\begin{equation} \label{eq:h-localh}
  h (\Gamma, x) \ = \sum_{F \subseteq V}
  \ell_F (\Gamma_F, x).
\end{equation}

Stanley~\cite{Sta92} showed
that $\ell_V (\Gamma, x)$ has nonnegative and
symmetric coefficients, so that $x^n
\ell_V (\Gamma, 1/x) = \ell_V (\Gamma, x)$, for
every triangulation $\Gamma$ of $2^V$, and that it
has unimodal coefficients for every \emph{regular}
triangulation, meaning that $\Gamma$ can be
realized as the collection of projections on a
geometric simplex of the lower faces of a simplicial
polytope of one dimension higher.

The \emph{barycentric subdivision} of
a simplicial complex $\Delta$ is defined as the
simplicial complex $\sd(\Delta)$ on the vertex set
$\Delta \sm \{\varnothing\}$ whose faces are the chains
$F_0 \subsetneq F_1 \subsetneq \cdots \subsetneq F_k$
of nonempty faces of $\Delta$. The carrier of such a
chain is its top element $F_k$.
To describe the $h$-polynomial and local
$h$-polynomial of $\sd(\sigma_n)$, we need to recall
a few definitions from permutation enumeration. An
\emph{excedance} of a permutation $w \in \fS_n$ is
an index $i \in [n-1]$ such that $w(i) > i$. Let
$\exc(w)$ be the number of excedances of $w$.
The polynomial

\begin{equation*}
A_n(x) \ = \ \sum_{w \in \fS_n} x^{\exc(w)}
\end{equation*}

is called \emph{$n$th Eulerian polynomial}	$A_n(x)$;
see \cite[Section~1.4]{StaEC1} for more information
on this important concept. Similarly, the \emph{$n$th
derangement polynomial} is defined by the formula

\begin{equation*}
d_n(x) \ = \ \sum_{w \in \fD_n}x^{\exc(w)},
\end{equation*}

where $\fD_n$ the set of all derangements (permutations
without fixed points) in $\fS_n$. Then, $h(\sd(\sigma_n),
x) = A_n(x)$ and $\ell_V(\sd(\sigma_n), x) = d_n(x)$ for
every $n$ (see \cite[Section~2]{Sta92}), where $V$ is the
vertex set of $\sigma_n$.

Let $\fF = (f_\fF(i,j))$ be a triangular array of
nonnegative integers, defined for $0 \le i \le j$. A
triangulation $\Delta'$ of a simplicial complex $\Delta$
is called \emph{$\fF$-uniform} if for every
$(n-1)$-dimensional face $F \in \Delta$, the restriction
$\Delta'_F$ has exactly $f_\fF(k,n)$ faces of
dimension $k-1$ for all $0 \le k \le n$. The
barycentric subdivision is a prototypical example of an
$\fF$-uniform triangulation, for a suitable array $\fF$;
the antiprism triangulation is another. The class of
$\fF$-uniform triangulations was introduced and studied
in~\cite{Ath20+}. The $h$-polynomial and local
$h$-polynomial of an $\fF$-uniform triangulation of
an $(n-1)$-dimensional simplex depend only on $\fF$ and
$n$ and will be denoted by $h_\fF(\sigma_n, x)$ and
$\ell_\fF(\sigma_n, x)$, respectively.

\subsection{Polynomials}
\label{sec:polys}

We recall some basic definitions and useful facts about
unimodal and real-rooted polynomials. A polynomial $p(x)
= a_0 + a_1 x + \cdots + a_n x^n \in \RR[x]$ is called

\begin{itemize}
\item[$\bullet$]
  \emph{symmetric}, with center of symmetry $n/2$, if
	$a_i = a_{n-i}$ for all $0 \le i \le n$,
\item[$\bullet$]
  \emph{unimodal}, with a peak at position $k$, if $a_0 \le
	 a_1 \le \cdots \le a_k \ge a_{k+1} \ge \cdots \ge a_n$,
\item[$\bullet$]
  \emph{alternatingly increasing} with respect to $n$, if
	 $a_0 \le a_n \le	a_1 \le a_{n-1} \le \cdots \le
	 a_{\lceil n/2 \rceil}$,
\item[$\bullet$]
  \emph{$\gamma$-positive}, with center of symmetry $n/2$,
	 if $p(x) = \sum_{j=0}^{\lfloor n/2 \rfloor} \gamma_j
	 x^j (1+x)^{n-2j}$ for some nonnegative reals $\gamma_0,
   \gamma_1,\dots,\gamma_{\lfloor n/2 \rfloor}$.
\end{itemize}

Gamma-positivity implies palindromicity and unimodality;
see \cite{Ath18} for a survey about this very
interesting concept.

A polynomial $p(x) \in \RR[x]$ is \emph{real-rooted}
if all complex roots of $p(x)$ are real, or $p(x)$ is
the zero polynomial. A real-rooted polynomial, with
roots $\alpha_1 \ge \alpha_2 \ge \cdots$, is said to
\emph{interlace} another real-rooted polynomial, with
roots $\beta_1 \ge \beta_2 \ge \cdots$, if

\[ \cdots \le \alpha_2 \le \beta_2 \le \alpha_1 \le
   \beta_1. \]

By convention, the zero polynomial interlaces and is
interlaced by every real-rooted polynomial and constant
polynomials interlace all polynomials of degree at most
one. Background on real-rooted polynomials and the theory
of interlacing can be found in \cite{Bra15, Fi06, Sta89}
and references therein. We recall here the crucial facts
that every real-rooted polynomial with nonnegative
coefficients is unimodal and that (see
\cite[Lemma~3.4]{Fi06}) if two real-rooted polynomials
$p(x)$ and $q(x)$ have positive leading coefficients and
$p(x)$ interlaces $q(x)$, then $p(x) + q(x)$ is
real-rooted as well and it is interlaced by $p(x)$ and
interlaces $q(x)$. Moreover, every symmetric real-rooted
polynomial is $\gamma$-positive.

A sequence $(p_0(x), p_1(x),\dots,p_m(x))$ of
real-rooted polynomials is called \emph{interlacing} if
$p_i(x)$ interlaces $p_j(x)$ for $0 \le i < j \le m$.
The following lemma will be used for the proof of
Theorem~\ref{thm:mainA} in Section~\ref{sec:enu-simplex}.

\begin{lemma} \label{lem:inter}

\begin{itemize}
\itemsep=0pt
\item[(a)]
{\rm (\cite[Lemma~2.3]{Bra06},
\cite[Proposition~3.3]{Wa92})}
Let $p_1(x), p_2(x),\dots,p_m(x)$ be real-rooted
polynomials in $\RR[x]$. If $p_1(x)$ interlaces
$p_m(x)$ and $p_i(x)$ interlaces $p_{i+1}(x)$ for
all $i \in [n-1]$, then $(p_1(x), p_2(x),\dots,p_m(x))$
is an interlacing sequence.

\item[(b)]
{\rm (cf. \cite[Lemma~3.4]{Fi06})}
If $(p_1(x), p_2(x),\dots,p_m(x))$ is an interlacing
sequence of real-rooted polynomials in $\RR[x]$ with
positive leading coefficients, then so is $(p_1(x) +
p_2(x) + \cdots + p_m(x),\dots,p_{m-1}(x) + p_m(x),
p_m(x))$.

\item[(c)]
Let $(p_1(x), p_2(x),\dots,p_m(x))$ be an interlacing
sequence of real-rooted polynomials in $\RR[x]$ with
positive leading coefficients. Then, $p_1(x) + p_2(x)
+ \cdots + p_m(x)$ interlaces $c_1 p_1(x) + c_2 p_2(x)
+ \cdots + c_m p_m(x)$ for all positive reals $c_1 \le
c_2 \le \cdots \le c_m$.
In particular, $p_1(x) + p_2(x) + \cdots + p_{m-1}(x)$
interlaces $p_1(x) + 2p_2(x) + \cdots + mp_m(x)$.
\end{itemize}
\end{lemma}

\begin{proof}
We only need to prove part (c) and for that, we proceed
by induction on $m$. The case $m=1$ being trivial, let
us assume that the result holds for a positive integer
$m-1$, consider a sequence $(p_1(x),
p_2(x),\dots,p_m(x))$ and positive reals $c_1 \le c_2
\le \cdots \le c_m$ as in the statement of the lemma
and set $s_m(x) := p_1(x) + p_2(x) + \cdots + p_m(x)$.
Since the sequence $(p_1(x),\dots,p_{m-2}(x),p_{m-1} +
p_m(x))$ is also interlacing \cite[Lemma~3.4]{Fi06}, the
induction hypothesis implies that $s_m(x)$ interlaces
$c_1 p_1(x) + \cdots + c_{m-2} p_{m-2}(x) + c_{m-1}
(p_{m-1}(x) + p_m(x))$. Since $s_m(x)$ also interlaces
$(c_m - c_{m-1})p_m(x)$ (because each of its summands
does so), it must interlace the sum of these two
polynomials. This completes the induction.

For the second statement, let $s_{m-1}(x) := p_1(x) +
p_2(x) + \cdots + p_{m-1}(x)$. From the first statement
we have that $s_{m-1}(x)$ interlaces $p_1(x) + 2p_2(x)
+ \cdots + (m-1)p_{m-1}(x)$. Since $s_{m-1}(x)$ also
interlaces $mp_m(x)$, it must interlace the sum of
these two polynomials and the proof follows.
\end{proof}

Every polynomial $p(x) \in \RR[x]$ of degree at most
$n$ can be written uniquely in the form $p(x) = a(x) +
xb(x)$, where $a(x)$ and $b(x)$ are symmetric with
centers of symmetry $n/2$ and $(n-1)/2$, respectively.
We say that $p(x)$ has a \emph{nonnegative symmetric
decomposition} with respect to $n$, if $a(x)$ and
$b(x)$ have nonnegative coefficients. Following
\cite{BS20}, we also say that $p(x)$ has a
\emph{real-rooted symmetric decomposition}
(respectively, \emph{real-rooted and interlacing
symmetric decomposition}) with respect to $n$, if
$a(x)$ and $b(x)$ are real-rooted (respectively, if
$a(x)$ and $b(x)$ are real-rooted and $x^n p(1/x)$
interlaces $p(x)$). By \cite[Theorem~2.6]{BS20}, if
$p(x)$ has a nonnegative, real-rooted and interlacing
symmetric decomposition with respect to $n$, then
$b(x)$ interlaces $a(x)$ and each one of them interlaces
$p(x)$. The alternatingly increasing property for $p(x)$,
defined earlier, with respect to $n$ is equivalent to
the unimodality of both $a(x)$ and $b(x)$.

\section{The antiprism construction}
\label{sec:aconstr}

The antiprism triangulation of a simplicial complex
can be defined geometrically by iterating the
antiprism construction. This section reviews the
latter and studies its face enumeration, in the
framework of uniform triangulations \cite{Ath20+}.
The results will be applied in
Section~\ref{sec:enumerate}, but may be of
independent interest too.

Let $V = \{v_1, v_2,\dots,v_n\}$ be an $n$-element
set and $\Delta$ be a triangulation of the boundary
complex of the simplex $2^V$. We pick an $n$-element
set $U = \{u_1, u_2,\dots,u_n\}$ which is disjoint
from the vertex set of $\Delta$ and denote by
$\Gamma_\aA(\Delta)$ the collection of faces of
$\Delta$ together with all sets of the form $E \cup
G$, where $E = \{ u_i~:~ i \in I\}$ is a nonempty face
of the simplex $2^U$ for some $\varnothing \subsetneq
I \subseteq [n]$ and $G$ is a face of the restriction
of $\Delta$ to the face $F = \{ v_i~:~ i \in [n] \sm I\}$
of $\partial (2^V)$ which is complementary to $E$.
The collection $\Gamma_\aA(\Delta)$ is a simplicial
complex which contains $2^U$ and $\Delta$ as
subcomplexes; we call it the \emph{antiprism over
$\Delta$}. When $\Delta = \partial(2^V)$ is the
trivial triangulation, the antiprism $\Gamma_\aA
(\partial(2^V))$ is combinatorially isomorphic to
the Schlegel diagram \cite[Section~5.2]{Zie95} of
the $n$-dimensional cross-polytope behind any of its
facets. For general $\Delta$, the antiprism
$\Gamma_\aA(\Delta)$ is a triangulation of
$\Gamma_\aA(\partial(2^V))$: the carrier of a face
$E \cup G$, as above, is the union of $E$ with the
carrier of $G$, the latter considered as a face of
the triangulation $\Delta$ of $\partial(2^V)$. Since
$\Gamma_\aA(\partial(2^V))$ triangulates the simplex
$2^V$, $\Gamma_\aA(\Delta)$ is a triangulation of
$2^V$ as well with boundary equal to $\Delta$.

\begin{remark} \label{rem:DeltaA} \rm
Given a triangulation $\Gamma$ of the
$(n-1)$-dimensional simplex $2^V$, an analogous
procedure defines a triangulation, say $\Delta_\aA
(\Gamma)$, of the $(n-1)$-dimensional sphere which
contains $2^U$ and $\Gamma$ as subcomplexes
and which we may call the \emph{antiprism over
$\Gamma$}. This construction was employed in
\cite[Section~4]{Ath12}, in order to relate the
$\gamma$-vector of a flag triangulation of the
sphere to the local $\gamma$-vector of a flag
triangulation of the simplex, and
in~\cite[Section~4]{Ath20}, in order to interpret
geometrically binomial Eulerian polynomials (see
Example~\ref{ex:binom-euler}) and
certain analogues for $r$-colored permutations.
The connection between the two constructions is
that $\Delta_\aA(\Gamma) = \Gamma \cup \Gamma_\aA
(\partial \Gamma)$.
\qed
\end{remark}

The following statement is closely related to
\cite[Proposition~4.1]{Ath20}.

\begin{proposition} \label{prop:A(Delta)}
The simplicial complex $\Gamma_\aA(\Delta)$
triangulates the $(n-1)$-dimensional simplex $2^V$
for every triangulation $\Delta$ of the boundary
complex $\partial(2^V)$. Moreover,

\[ h(\Gamma_\aA(\Delta), x) \ = \ \sum_{F \subsetneq
   V} x^{|F|} h(\Delta_F, 1/x). \]

\end{proposition}

\begin{proof}
We have already commented on the first sentence.
For the second, using
Proposition~\ref{prop:h-ballcirc} and the
definition of the $h$-polynomial we find that

\begin{eqnarray*}
x^n h(\Gamma_\aA(\Delta), 1/x) & = &
h^\circ(\Gamma_\aA(\Delta), x) \ = \
\sum_{G \in \Gamma_\aA(\Delta)^\circ} x^{|G|}
(1-x)^{n-|G|} \\ & = &
\sum_{\varnothing \ne E \subseteq U}
\sum_{\scriptsize \begin{array}{c} G \in
\Gamma_\aA(\Delta),
\\ G \cap U = E \end{array}} x^{|G|} (1-x)^{n-|G|}.
\end{eqnarray*}

By definition of $\Gamma_\aA(\Delta)$, the inner
sum is equal to $x^{|E|} h(\Delta_F, x)$, where
$F \subsetneq V$ is the face of $2^V$ which is
complementary to $E$. Replacing $x$ by $1/x$ results
in the proposed expression for $h(\Gamma_\aA(\Delta),
x)$ and the proof follows.
\end{proof}

We now turn our attention to uniform triangulations
of $\partial (2^V)$.

\begin{proposition} \label{prop:A(uDelta)}
For every $\fF$-uniform triangulation $\Delta$ of
the boundary complex of an $(n-1)$-dimensional
simplex $2^V$:

\begin{eqnarray}
h(\Gamma_\aA(\Delta), x) & = & \sum_{k=0}^{n-1}
   {n \choose k} x^k h_\fF(\sigma_k, 1/x)
	          \label{eq:h-A(Delta)a} \\
& = & \sum_{k=0}^{n-1} {n \choose k}
	  \ell_\fF(\sigma_k, x) \left( (1+x)^{n-k} - x^{n-k}
		\right) \label{eq:h-A(Delta)b}, \\
\ell_V(\Gamma_\aA(\Delta), x) & = & \sum_{k=0}^{n-1}
   {n \choose k} \ell_\fF(\sigma_k, x)
	 \left( (1+x)^{n-k} - 1 - x^{n-k} \right)
	         \label{eq:ell-A(Delta)}, \\
h(\Gamma_\aA(\Delta), x) - h(\Delta, x)
& = & \sum_{k=0}^{n-1} {n \choose k} \ell_\fF
  (\sigma_k, x) \left( (1+x)^{n-k} - 1 - x -
	\cdots - x^{n-k} \right) \label{eq:h-hbd-A(Delta)}
\\ & = & \sum_{k=0}^{n-1}
   {n \choose k} h_\fF(\sigma_k, x)
	 \left( x^{n-k} - x (x-1)^{n-k-1} \right)	.
	         \label{eq:h-hbd-h-A(Delta)}
\end{eqnarray}

\medskip
In particular, if all restrictions of $\Delta$ to
proper faces of $2^V$ are regular triangulations,
then the polynomials $\ell_V(\Gamma_\aA(\Delta), x)$ and
$h(\Gamma_\aA(\Delta), x) - h(\Delta, x)$ are
unimodal and $h(\Gamma_\aA(\Delta), x)$ is alternatingly
increasing with respect to $n-1$.

\end{proposition}

\begin{proof}
Equation~(\ref{eq:h-A(Delta)a}) follows directly
from Proposition~\ref{prop:A(Delta)}. To deduce
Equation~(\ref{eq:h-A(Delta)b}) from that, we use
(\ref{eq:h-localh}) to express $h_\fF(\sigma_k, 1/x)$
in terms of local $h$-polynomials, apply the symmetry
property of the latter and change the order of
summation to obtain

\begin{eqnarray*}
h(\Gamma_\aA(\Delta), x) & = & \sum_{k=0}^{n-1}
   {n \choose k} x^k \sum_{j=0}^k {k \choose j}
	 \ell_\fF(\sigma_j, 1/x) \ = \
	\sum_{k=0}^{n-1} {n \choose k} x^{k-j} \sum_{j=0}^k
	{k \choose j} \ell_\fF(\sigma_j, x) \\
& = & \sum_{j=0}^{n-1} \ell_\fF(\sigma_j, x)
  \sum_{k=j}^{n-1} {n \choose k} {k \choose j} x^{k-j}
\ = \ \sum_{j=0}^{n-1} {n \choose j} \ell_\fF(\sigma_j, x)
  \sum_{k=j}^{n-1} {n-j \choose n-k} x^{k-j} \\ & = &
	\sum_{j=0}^{n-1} {n \choose j}
	  \ell_\fF(\sigma_j, x) \left( (1+x)^{n-j} - x^{n-j}
		\right).
\end{eqnarray*}

For the fourth and fifth step we have used the identity
${n \choose k} {k \choose j} = {n \choose j} {n-j
\choose n-k}$ and the binomial theorem, respectively.

Alternatively,
Equation~(\ref{eq:h-A(Delta)b}) follows from an
application of Stanley's locality formula
\cite[Theorem~3.2]{Sta92} to $\Gamma_\aA(\Delta)$,
considered as a triangulation of the antiprism
$\Gamma_\aA(\partial (2^V))$ over the boundary
complex of $2^V$. Equation~(\ref{eq:ell-A(Delta)})
follows when combining (\ref{eq:h-A(Delta)b}) with

\begin{equation} \label{eq:h-ell-sigma}
  h(\Gamma_\aA(\Delta), x) \ = \
	\ell_V(\Gamma_\aA(\Delta), x)
	\, + \, \sum_{k=0}^{n-1} {n \choose k}
  \ell_\fF(\sigma_k, x),
\end{equation}

the latter being (\ref{eq:h-localh}) applied to
$\Gamma_\aA(\Delta)$. Equation~(\ref{eq:h-hbd-A(Delta)})
follows from~(\ref{eq:h-A(Delta)b}) and

\begin{equation} \label{eq:h-bdsigma}
 h(\Delta, x) \ = \ \sum_{k=0}^{n-1} {n \choose k}
   \ell_\fF(\sigma_k, x)
   (1 + x + x^2 + \cdots + x^{n-k-1}),
\end{equation}

which is also a consequence of \cite[Theorem~3.2]
{Sta92}; see \cite[Equation~(4.2)]{KMS19}.
Equation~(\ref{eq:h-hbd-h-A(Delta)}) follows
from~(\ref{eq:h-hbd-A(Delta)}) by expressing $\ell_\fF
(\sigma_k, x)$ in terms of the $h$-polynomials
$h_\fF(\sigma_j, x)$, changing the order of
summation and computing the inner sum, just as
in the proof of Equation~(\ref{eq:h-A(Delta)b}); we
leave the details of this computation to the
interested reader.

For the last statement we
note that, by the regularity assumption, $\ell_\fF
(\sigma_k, x)$ is (symmetric with center of symmetry
$k/2$ and) unimodal for $0 \le k < n$. As a
result, Equations~(\ref{eq:ell-A(Delta)})
and~(\ref{eq:h-hbd-A(Delta)}) imply the unimodality
of $\ell_V(\Gamma_\aA(\Delta), x)$ and
$h(\Gamma_\aA(\Delta), x) - h(\Delta, x)$,
respectively, and Equations~(\ref{eq:h-hbd-A(Delta)})
and~(\ref{eq:h-bdsigma}) imply that the symmetric
decomposition

\[ h(\Gamma_\aA(\Delta), x) \ = \ h(\Delta, x) +
   (h(\Gamma_\aA(\Delta), x) - h(\Delta, x)) \]

of $h(\Gamma_\aA(\Delta), x)$ with respect to $n-1$
is nonnegative and unimodal. The latter statement is
	equivalent to $h(\Gamma_\aA(\Delta),x)$ being alternatingly
	increasing.
\end{proof}

\begin{remark} \label{rem:c(Delta)} \rm
Let $\Delta$ be as in
Proposition~\ref{prop:A(uDelta)}. Since coning
a simplicial complex does not affect the
$h$-polynomial, the right-hand side
of~(\ref{eq:h-bdsigma}) is also an expression for
$h(u \ast\Delta, x)$, where $u \ast \Delta$ denotes
the cone of $\Delta$ with apex $u$. The formula

\begin{equation} \label{eq:h-bdsigma-new}
  h(u \ast\Delta, x) \ = \ \sum_{k=0}^{n-1}
  {n \choose k} h_\fF(\sigma_k, x) (x-1)^{n-k-1}
\end{equation}

can be derived from that by expressing $\ell_\fF
(\sigma_k, x)$ in terms of the $h$-polynomials
$h_\fF(\sigma_j, x)$, changing the order of
summation and computing the inner sum, just as
in the proof of Equations~\eqref{eq:h-A(Delta)b}
and \eqref{eq:h-hbd-h-A(Delta)}
or, alternatively, by adapting the argument in
the proof of Proposition~\ref{prop:A(Delta)}.
When $\Delta$ is the barycentric subdivision of
$\partial \sigma_n$, this yields the recursion

\[ A_n(x) \ = \ \sum_{k=0}^{n-1} {n \choose k}
   A_k(x) (x-1)^{n-k-1} \]

for the Eulerian polynomial $A_n(x)$, valid for
$n \ge 1$. This appears as Equation~(2.7) in \cite{Fo08}.

\end{remark}

\begin{example} \label{ex:binom-euler} \rm
Suppose again that $\Delta$ is the barycentric
subdivision of $\partial \sigma_n$. Then,
Equation~(\ref{eq:h-A(Delta)a}) yields that

\[ h(\Gamma_\aA(\Delta), x) \ = \ \sum_{k=0}^{n-1}
   {n \choose k} x^k A_k(1/x) \ = \ 1 \, + \,
	 x \sum_{k=1}^{n-1} {n \choose k} A_k(x)
	 \ = \ \widetilde{A}_n(x) - xA_n(x) \]

and $h(\Gamma_\aA(\Delta), x) - h_\fF(\partial \sigma_n)
= \widetilde{A}_n(x) - (1+x)A_n(x)$, where

\[ \widetilde{A}_n(x) \ := \ 1 \, + \, x
   \sum_{k=1}^n {n \choose k} A_k(x) \]

is the $n$th binomial Eulerian polynomial studied,
for instance, in \cite{Ath20, SW20}. From
Equation~(\ref{eq:h-ell-sigma}) we compute
further that $\ell_V(\Gamma_\aA(\Delta), x) =
\widetilde{A}_n(x) - (1+x)A_n(x) - d_n(x)$,
where $d_n(x) = \ell_\fF(\sigma_n, x)$ is the
$n$th derangement polynomial (see
Section~\ref{sec:triang}).

Therefore, by Proposition~\ref{prop:A(uDelta)},
$\widetilde{A}_n(x) - xA_n(x)$ is alternatingly
increasing with respect to $n-1$ and
$\widetilde{A}_n(x) - (1+x)A_n(x)$ is symmetric
and unimodal.
\qed
\end{example}

\section{The antiprism triangulation}
\label{sec:atriang}

This section briefly describes combinatorially
and geometrically the antiprism triangulation of a
simplicial complex. For more information we refer
to \cite[Apendix~A.1]{IJ03} and
\cite{Ko12}, where these descriptions are given in
variant forms. We first review the corresponding
descriptions of the barycentric subdivision, which we
will parallel to treat the antiprism triangulation.

Let $\Delta$ be a simplicial
complex. Consider the (simple, undirected) graph
$\gG(\Delta)$ on the node set of nonempty faces
of $\Delta$ for which two nodes are adjacent if one
is contained in the other. The barycentric subdivision
$\sd(\Delta)$ is defined as the \emph{clique complex}
of $\gG(\Delta)$, meaning the abstract simplicial
complex whose vertices are the nodes of $\gG(\Delta)$
and whose faces are the sets consisting of pairwise
adjacent nodes. This is equivalent to the
definition already given in Section~\ref{sec:triang}.

Geometrically, $\sd(\Delta)$
can be described as a triangulation of $\Delta$ as
follows. Assume that all faces of $\Delta$ of
dimension at most $j$ have been triangulated, for
some $j \in \NN$. Then, triangulate each
$(j+1)$-dimensional face of $\Delta$ by inserting
one point in the interior of that face and coning
over its boundary, which is already triangulated.
By repeating this process, starting at $j=0$ and
moving to higher dimensional faces, we get a
triangulation of $\Delta$ which is
combinatorially isomorphic to $\sd(\Delta)$.
Alternatively, $\sd(\Delta)$ can be constructed
by applying successively the operation of stellar
subdivision to each face of $\Delta$ of positive
dimension, starting from the facets and moving to
lower dimensional faces in any order which respects
reverse inclusion. A \emph{stellar subdivision} on
a face $F \in \Delta$ replaces $\Star_\Delta(F)$ by
the join of $\link_\Delta(F)$ with the cone over
$\partial(2^F)$.

The antiprism triangulation can be defined
similarly, if the nonempty faces of $\Delta$ are
replaced by pointed faces and coning is replaced by
the antiprism construction of
Section~\ref{sec:aconstr}. Recall that a
\emph{pointed subset} of a set $V$ is any pair
$(S, v)$ such that $v \in S \subseteq V$. Similarly,
a \emph{pointed face} of a simplicial complex $\Delta$
is any pair $(F,v)$ such that $F \in \Delta$ is a
face and $v \in F$ is a chosen vertex.

\begin{definition} \label{def:sdA-complex}
Let $\Delta$ be a simplicial
complex. We denote by $\gG_\aA(\Delta)$ the (simple,
undirected) graph on the node set of pointed faces
of $\Delta$ for which two distinct pointed faces
$(F,v)$ and $(F',v')$ are adjacent if

\begin{itemize}
\itemsep=0pt
\item[$\bullet$]
$F = F'$, or

\item[$\bullet$]
$F \subsetneq F'$ and $v' \in (F' \sm F)$, or

\item[$\bullet$]
$F' \subsetneq F$ and $v \in (F \sm F')$.
\end{itemize}

The \emph{antiprism triangulation} of $\Delta$, denoted
by $\sd_\aA(\Delta)$, is the abstract simplicial
complex defined as the clique complex of $\gG_\aA
(\Delta)$.
\end{definition}

Examples of antiprism triangulations are shown in
Figures~\ref{fig:2simplex} and~\ref{fig:Bsp2}.

\begin{figure}
\centering
  \includegraphics[width=0.7\textwidth]{Bsp2}
  \caption{Antiprism triangulation of the cone
	over the boundary of the 2-simplex}
  \label{fig:Bsp2}
\end{figure}

The faces of $\sd_\aA(\Delta)$ can be described
explicitly, in combinatorial terms
\cite[Section~2]{Ko12}. Given a set $S$, an
\emph{ordered set partition} (or simply, ordered
partition) of $S$ is any sequence of nonempty,
pairwise disjoint sets (called \emph{blocks}) whose
union is equal to $S$. A \emph{multi-pointed
ordered partition} of $S$ is defined as a pair
$(\pi, \tau)$, where $\pi = (B_1, B_2,\dots,B_m)$
and $\tau = (C_1, C_2,\dots,C_m)$ are ordered
partitions of $S$ and of a subset of $S$,
respectively, with the same number of blocks,
such that $C_i$ is a nonempty subset of $B_i$ for
every $i \in [m]$. We think of such a pair as an
ordered partition of $S$, together with a choice
of a nonempty subset for every block. The sum of the
cardinalities of these subsets $C_i$ (total number
of chosen elements) will be called the \emph{weight}
of $(\pi, \tau)$. Then, the $(k-1)$-dimensional
faces of $\sd_\aA(\Delta)$ are in one-to-one
correspondence with the multi-pointed ordered
partitions of faces of $\Delta$ of weight $k$.
More specifically, the multi-pointed ordered
partition $(\pi, \tau)$, with $\pi = (B_1,
B_2,\dots,B_m)$ and $\tau = (C_1, C_2,\dots,C_m)$,
corresponds to the face of $\sd_\aA(\Delta)$ with
vertices the pointed faces $(F, v)$ of $\Delta$,
where $F = B_1 \cup B_2 \cup \cdots \cup B_i$ for
some $i \in [m]$ and $v \in C_i$.
The faces of the antiprism triangulation of the simplex
$2^V$ are the multi-pointed ordered partitions of
subsets of $V$; they will be referred to as
\emph{multi-pointed partial ordered partitions} of
$V$. Note that the facets of $\sd_\aA(\Delta)$ are
in one-to-one correspondence with the ordered
partitions of the facets of $\Delta$ (since all
elements in the blocks should be chosen).
Figure \ref{fig:2simplex} shows the antiprism
triangulation of the 2-simplex, including some faces
labeled by multi-pointed ordered partitions.

As was the case with barycentric subdivision,
$\sd_\aA(\Delta)$ can be constructed geometrically
by applying the antiprism construction of
Section~\ref{sec:aconstr} to its faces, starting
from the edges and moving to faces of higher
dimension in any order which respects inclusion.
This process is slightly different from the one
in \cite{IJ03, Ko12} which uses crossing operations
on the faces of $\Delta$ instead, starting from
facets and moving to faces of lower dimension in
any order which respects reverse inclusion. A
\emph{crossing operation} (also known as a
\emph{balanced stellar subdivision} \cite{BM87})
on a face $F \in \Delta$ replaces $\Star_\Delta(F)$
by the join of $\link_\Delta(F)$ with the antiprism
(as defined in Section~\ref{sec:aconstr}) over
$\partial(2^F)$. Both approaches result in a
triangulation which is combinatorially isomorphic
to $\sd_\aA(\Delta)$. Under this isomorphism,
the carrier of a multi-pointed ordered partition
of a face $F \in \Delta$ is equal to $F$. As a
result, the interior faces of the antiprism
triangulation of the simplex $2^F$ are in
one-to-one correspondence with the multi-pointed
ordered partitions of $F$.

\section{Face enumeration}
\label{sec:enumerate}

This section studies the rich enumerative
combinatorics of antiprism triangulations and
proves Theorem~\ref{thm:mainA}. Following the
notation of \cite{Ath20+}, we denote by
$h_\aA(\sigma_n, x)$ and $\ell_\aA(\sigma_n, x)$
the $h$-polynomial and local $h$-polynomial of
$\sd_\aA(\sigma_n)$, respectively. These two
polynomials play an important role in this study.
The main difficulty for proving the real-rootedness
of $h_\aA(\sigma_n, x)$ comes from the fact that
we know of no simpler recurrence relation for it
than that of \Cref{prop:h-sdA(simplex)}. Some
of the combinatorial interpretations of $h_\aA
(\sigma_n,x)$ extend to describe the effect of
the antiprism triangulation on the $h$-polynomial
of any simplicial complex.

\subsection{The antiprism triangulation of a
simplex}
\label{sec:enu-simplex}

As discussed in Section~\ref{sec:atriang}, the
number of $(k-1)$-dimensional faces of the
antiprism triangulation $\sd_\aA(\sigma_n)$ is
equal to the number of multi-pointed partial
ordered set partitions of $[n]$ of weight $k$.
We now give a recurrence and combinatorial
interpretations for the $h$-polynomial of
$\sd_\aA(\sigma_n)$. For the first few values
of $n$,

\[  h_\aA(\sigma_n, x) \ = \ \begin{cases}
    1, & \text{if $n = 0$} \\
		1, & \text{if $n = 1$} \\
    1 + 2x, & \text{if $n=2$} \\
    1 + 9x + 3x^2, & \text{if $n=3$} \\
    1 + 28x + 42x^2 + 4x^3, & \text{if $n=4$} \\
    1 + 75x + 310x^2 + 150x^3 + 5x^4, &
                            \text{if $n=5$} \\
    1 + 186x + 1725x^2 + 2300x^3 + 465x^4 + 6x^5,
                           & \text{if $n=6$} \\
		1 + 441x + 8211x^2 + 23625x^3 + 13685x^4 +
		    1323x^5 + 7x^6, & \text{if $n=7$.}
    \end{cases}  \]

We first need to introduce some more terminology.
Let $\varphi = (\pi, \tau)$ be a multi-pointed
partial ordered set partition of $[n]$. Thus, $\pi
= (B_1, B_2,\dots,B_m)$ is an ordered partition of
a subset $S$ of $[n]$ and $\tau = (C_1,
C_2,\dots,C_m)$, where $C_i$ is a nonempty subset
of $B_i$ for every $i \in [m]$. We will say that
$\varphi$ is \emph{proper} if $C_i$ is a proper
subset of $B_i$ for every $i \in [m]$. We will
use the same terminology with the adjective
`partial' dropped, when $S = [n]$. The
\emph{excedance set} of a permutation $w \in \fS_n$
is defined as the set of indices $i \in [n-1]$ such
that $w(i) > i$; see \cite{ES00} for more
information on this concept.

\begin{proposition} \label{prop:h-sdA(simplex)}

\begin{itemize}
\itemsep=0pt
\item[{\rm (a)}]
We have

\begin{equation} \label{eq:hA(simplex)-rec}
h_\aA(\sigma_n, x) \ = \ \sum_{k=0}^{n-1}
{n \choose k} x^k h_\aA(\sigma_k, 1/x)
\end{equation}

for every positive integer $n$.

\item[{\rm (b)}]
The coefficient of $x^k$ in $h_\aA(\sigma_n, x)$
is equal to:

\begin{itemize}
\itemsep=0pt
\item[$\bullet$]
the number of proper multi-pointed partial ordered
set partitions of $[n]$ of weight $k$,

\item[$\bullet$]
the number of ways to choose a subset $S \subseteq
[n]$ and an ordered set partition $\pi$ of $S$ and
to color $k$ elements of $S$ black and the remaining
elements white, so that no block of $\pi$ is
monochromatic,

\item[$\bullet$]
the number of ordered set partitions $\pi =
(B_1, B_2,\dots,B_m)$ of $[n]$ for which the
union $\bigcup_{i=1}^{\lfloor m/2 \rfloor} B_i$
has exactly $k$ elements,

\item[$\bullet$]
${n \choose k}$ times the number of permutations
in $\fS_n$ with excedance set equal to $[k]$,

\item[$\bullet$]
the explicit expression $$ {n \choose k}
\sum_{j=1}^{k+1} \, (-1)^{k+1-j} j! S(k+1,j)
j^{n-k-1}, $$
\end{itemize}

where $S(n,k)$ are the Stirling numbers of the
second kind.
\end{itemize}

\end{proposition}

\begin{proof}
Part (a) follows from
Proposition~\ref{prop:A(uDelta)}, as a special case
of Equation~(\ref{eq:h-A(Delta)a}). For part (b),
we first note that from Equation \eqref{eq:ell-A(Delta)}
of the same proposition and
Equation~(\ref{eq:h-localh}) we get

\[ \ell_\aA(\sigma_n, x) \ = \ \sum_{m=0}^{n-1}
   {n \choose m} \ell_\aA(\sigma_m, x)
	 \left( (1+x)^{n-m} - 1 - x^{n-m} \right)	\]

for $n \ge 1$ and

\[ h_\aA(\sigma_n, x) \ = \ \sum_{m=0}^n
   {n \choose m} \ell_\aA(\sigma_m, x),	\]

respectively. By induction on $n$, the former
equality implies that the coefficient of $x^k$ in
$\ell_\aA(\sigma_n, x)$ is equal to the number of
proper multi-pointed ordered set partitions of
$[n]$ of weight $k$. This and the latter equation
yield the first interpretation of $h_\aA(\sigma_n,
x)$ claimed in part (b). The second interpretation
is a restatement of the first (where black elements
correspond to the chosen elements in the blocks of
the multi-pointed partition).

The third interpretation can be deduced from the
first as follows. Let $Q(n,k)$ denote the collection
of proper multi-pointed partial ordered partitions
of $[n]$ of weight $k$. Each element of $Q(n,k)$
is a triple consisting of a subset $S \subseteq [n]$,
an ordered partition $\pi = (B_1, B_2,\dots,B_r)$
of $S$ and a choice of nonempty proper subset $C_i$
of $B_i$ for every $i \in [r]$,
such that the union $\cup_{i=1}^r C_i$ has
cardinality $k$. From such a triple one can define
an ordered partition of $[n]$ by listing the blocks
$C_1,\dots,C_r, B_1 \sm C_1,\dots,B_r \sm C_r$ in
this order and, if nonempty, adding $[n] \sm S$
at the end as the last block. It is straightforward
to verify that the resulting map is a bijection
from $Q(n,k)$ to the collection of ordered partitions
of $[n]$ described in the third proposed
interpretation.

For the last two claimed interpretations, let us
denote by $c(n,k)$ the number of permutations in
$\fS_n$ with excedance set equal to $[k]$,
for $k \in \{0, 1,\dots,n\}$. Then, $c(n,n) = 0$
and, as a consequence of Lemma~2.2 and Theorem~2.5
in \cite{ES00} (see also Section~3 of this
reference), $c(n,k) = c(n,n-k-1)$ and

\[ c(n, k) \ = \ 1 \, + \sum_{m=1}^k
   {k+1 \choose m} c(n-k-1+m, m). \]

for $k \in \{0, 1,\dots,n-1\}$. In view of $c(n,k)
= c(n,n-k-1)$, the latter equality can be
rewritten as

\begin{equation} \label{eq:c(n,k)-rec}
c(n, k) \ = \ 1 \, + \sum_{m=1}^{n-k-1} {n-k
\choose m} c(k+m, m).
\end{equation}

On the other hand, writing $h_\aA(\sigma_n, x) =
\sum_{k=0}^n p_\aA(n,k) x^k$ for $n \in \NN$, the
recursion of part (a) gives that

\[ p_\aA(n, k) \ = \ \sum_{m=k}^{n-1} {n \choose m}
   p_\aA(m, m-k) \]

for $k \in \{0, 1,\dots,n-1\}$. Setting $$ p_\aA(n,k)
\ = \ {n \choose k} \bar{p}_\aA(n,k), $$ the last
recursion can be rewritten as

\begin{eqnarray*}
{n \choose k} \bar{p}_\aA(n,k) & = &
\sum_{m=k}^{n-1} {n \choose m} {m \choose k}
\bar{p}_\aA(m, m-k) \ \ \text{i.e.,}\\
\bar{p}_\aA(n,k) & = & \sum_{m=k}^{n-1}
{n-k \choose m-k} \bar{p}_\aA(m, m-k) \ = \
\sum_{m=0}^{n-k-1} {n-k \choose m} \bar{p}_\aA(k+m, m).
\end{eqnarray*}

Comparing this recursion to~(\ref{eq:c(n,k)-rec})
we get that $\bar{p}_\aA(n, k) = c(n,k)$ for all $n$
and all $0\leq k\leq n$.
This proves the next to last interpretation, claimed
in part (b). The last interpretation follows from
this and the explicit formula for $c(n, k)$ obtained
in \cite[Proposition~6.5]{ES00}.
\end{proof}

The following statement is the main result of this
section.

\begin{theorem} \label{thm:h-sdA(simplex)-roots}
The polynomial $h_\aA(\sigma_n, x)$ is real-rooted
and interlaces $h_\aA(\sigma_{n+1}, x)$ for every
$n \in \NN$. Moreover, $h_\aA(\sigma_n, x)$ has a
nonnegative, real-rooted and interlacing symmetric
decomposition with respect to $n-1$, for every
positive integer $n$.
\end{theorem}

\begin{proof}
We consider the polynomials

\[ q_{n,r} (x) \ := \ \sum_{k=0}^n {n \choose k}
   x^{k+r} h_\aA(\sigma_{k+r}, 1/x), \]

shown in Table~\ref{tab:qnr} for the first few
values of $n, r \in \NN$. By part (a) of
Proposition~\ref{prop:h-sdA(simplex)} and the
definition of $q_{n,r}(x)$ we have

\begin{eqnarray}
q_{n,0} (x) & = & h_\aA(\sigma_n, x) +
         x^n h_\aA(\sigma_n, 1/x), \label{eq:qn0} \\
q_{0,r} (x) & = & x^r h_\aA(\sigma_r, 1/x)
                                   \label{eq:q0n}
\end{eqnarray}

for every positive integer $n$ and every $r \in \NN$,
respectively. We claim that

\[ \qQ_n \ := \ (q_{n,0}(x),
   q_{n-1,1}(x),\dots,q_{1,n-1}(x), q_{0,n}(x),
	 q_{0,n+1}(x)) \]

is an interlacing sequence of real-rooted polynomials
for every $n \in \NN$. In particular, selecting the
first and last two terms, we have the interlacing
sequence

\[ (h_\aA(\sigma_n, x) + x^n h_\aA(\sigma_n, 1/x),
    x^n h_\aA(\sigma_n, 1/x), x^{n+1}
		h_\aA(\sigma_{n+1}, 1/x)) \]

of real-rooted polynomials for every $n \in \NN$.
Before we prove the claim let us observe that,
since $x^n h_\aA(\sigma_n, 1/x)$ and $x^{n+1}
h_\aA(\sigma_{n+1}, 1/x)$ have degrees $n$ and
$n+1$, respectively, the
statement that the former polynomial interlaces
the latter is equivalent to the statement that
$h_\aA(\sigma_n, x)$ interlaces $h_\aA(\sigma_{n+1},
x)$. Similarly, since $h_\aA(\sigma_n, x) + x^n
h_\aA(\sigma_n, 1/x)$ is symmetric of degree $n$,
the statement that this polynomial interlaces
$x^{n+1} h_\aA(\sigma_{n+1}, 1/x)$ is equivalent
to each of the statements that the same polynomial
is interlaced by $x^n h_\aA(\sigma_{n+1}, 1/x)$
and that it interlaces $h_\aA(\sigma_{n+1}, x)$.

We now prove the claim by induction on $n$. This
is true for $n=0$, since $\qQ_0 = (1, x)$. We
assume that it holds for $n-1 \in \NN$. The
standard recurrence for the binomial coefficients
shows that $q_{n,r} (x) = q_{n-1,r} (x) +
q_{n-1,r+1}(x)$ for every $r \in \NN$. Writing
this in the form

\[ q_{n-r,r} (x) \ = \ q_{n-r-1,r} (x) +
                       q_{n-r-1,r+1} (x) \]

and iterating, we get

\begin{equation} \label{eq:q-rec}
q_{n-r,r} (x) \ = \ q_{n-r-1,r} (x) + q_{n-r-2,r+1}
     (x) + \cdots + q_{0,n-1} (x) + q_{0,n} (x)
\end{equation}

for $r \in \{0, 1,\dots, n\}$. This means that the
first $n+1$ terms of $\qQ_n$ are the partial sums
of the reverse of $\qQ_{n-1}$ and hence they form
an interlacing sequence, by part (b) of
Lemma~\ref{lem:inter}. Thus, by part (a) of this
lemma, to complete the induction it suffices
to show that $q_{n,0} (x)$ and $q_{0,n} (x)$
interlace $q_{0,n+1} (x)$. As already discussed,
and in view of~(\ref{eq:qn0}) and~(\ref{eq:q0n}),
this is equivalent to showing that
$h_\aA(\sigma_n, x) + x^n h_\aA(\sigma_n, 1/x)$
and $h_\aA(\sigma_n, x)$ interlace
$h_\aA(\sigma_{n+1}, x)$. To verify this we note
that, setting $r=0$ in Equation~(\ref{eq:q-rec}),
comparing with~(\ref{eq:qn0}) and~(\ref{eq:q0n})
and replacing $n$ with $n+1$, we get

\begin{equation}
q_{n,0} (x) + q_{n-1,1} (x) + \cdots + q_{0,n} (x)
\ = \ h_\aA(\sigma_{n+1}, x). \label{eq:qsum}
\end{equation}

Since the sum of the terms of an interlacing
sequence is interlaced by the first term, we
conclude that $h_\aA(\sigma_n, x) + x^n
h_\aA(\sigma_n, 1/x)$ interlaces
$h_\aA(\sigma_{n+1}, x)$. Finally, applying
part (c) of Lemma~\ref{lem:inter} to the
interlacing sequence $\qQ_{n-1}$ we conclude
that the sum of the first $n$ terms of this
sequence, which equals $h_\aA(\sigma_n, x)$,
interlaces the sum of the partial sums of the
reverse of $\qQ_{n-1}$, which equals
$h_\aA(\sigma_{n+1}, x)$. This completes the
proof of the claim.

Finally, note that $x^n h_\aA(\sigma_n, 1/x)$
and $x^{n+1} h_\aA(\sigma_{n+1}, 1/x)$ are the
last two terms of $\qQ_n$. Since this sequence
is interlacing, the two polynomials are
real-rooted and the former interlaces the
latter. As already discussed, this means that
$h_\aA(\sigma_n, x)$ is real-rooted and
interlaces $h_\aA(\sigma_{n+1}, x)$. Similarly,
the sum of the first $n$ terms of the sequence
$\qQ_{n-1}$ interlaces the last term. In view
of~(\ref{eq:q0n}) and~(\ref{eq:qsum}), this
means that $h_\aA(\sigma_n, x)$ interlaces
$x^n h_\aA(\sigma_n, 1/x)$ and, equivalently,
that $h_\aA(\sigma_n, x)$ is interlaced by
$x^{n-1} h_\aA(\sigma_n, 1/x)$. Since we already
know from Proposition~\ref{prop:A(uDelta)}
that $h_\aA(\sigma_n, x)$ has a nonnegative
symmetric decomposition with respect to $n-1$,
this decomposition must be real-rooted and
interlacing by \cite[Theorem~2.6]{BS20}.
\end{proof}

{\scriptsize
\begin{table}[hptb]
\begin{center}
\begin{tabular}{| l || l | l | l | l | l |} \hline
& $r=0$ & $r=1$ & $r=2$    & $r=3$ \\ \hline \hline
$n=0$   & 1    & $x$    & $2x+x^2$ & $3x+9x^2+x^3$ \\
         \hline
$n=1$   & $1+x$ & $3x+x^2$ & $5x+10x^2+x^3$ &
          $7x+51x^2+29x^3+x^4$ \\ \hline
 $n=2$  & $1+4x+x^2$ & $8x+11x^2+x^3$ &
          $12x+61x^2+30x^3+x^4$ & \\ \hline
 $n=3$  & $1+12x+12x^2+x^3$ & $20x+72x^2+31x^3+x^4$
        & & \\ \hline
\end{tabular}
\caption{Some polynomials $q_{n,r}(x)$.}
\label{tab:qnr}
\end{center}
\end{table}
}

Let us write $\theta_\aA(\sigma_n, x) := h_\aA
(\sigma_n, x) - h_\aA(\partial \sigma_n, x)$.
As mentioned in the proof of
Proposition~\ref{prop:A(uDelta)}, the expression
$h_\aA(\sigma_n, x) = h_\aA(\partial \sigma_n, x)
+ \theta_\aA(\sigma_n, x)$ is the symmetric
decomposition of $h_\aA(\sigma_n, x)$ with respect
to $n-1$. Thus, $h_\aA(\partial \sigma_n, x)$ and
$\theta_\aA(\sigma_n, x)$ are real-rooted by
Theorem~\ref{thm:h-sdA(simplex)-roots}. Although
the latter appears to be a very special case of
Conjecture~\ref{conj:main}, according to
\cite[Theorem~1.2]{Ath20+}, it would imply the
conjecture if the following statement (which we
have verified computationally for $n \le 20$)
also turns out to be true.

\begin{conjecture} \label{conj:thetaA}
The polynomial $h_\aA(\sigma_{n-1}, x)$ interlaces
$\theta_\aA(\sigma_n, x)$ for every positive
integer $n$.
\end{conjecture}

\begin{remark} \label{rem} \rm
The polynomial $h_\aA(\sigma_n, x) + x^n h_\aA
(\sigma_n, 1/x)$, shown to be real-rooted in the
proof of Theorem~\ref{thm:h-sdA(simplex)-roots},
is equal to the $h$-polynomial of a flag
triangulation of the $(n-1)$-dimensional sphere.
Indeed, let $\Gamma = \sd_\aA(\sigma_n)$, so that
$h(\Gamma, x) = h_\aA(\sigma_n, x)$. Then,
in the notation of Section~\ref{sec:aconstr}, in
particular Remark \ref{rem:DeltaA},
$\Delta = \Delta_\aA(\Gamma)$ is a flag
triangulation of the $(n-1)$-dimensional sphere
and $h(\Delta, x) = h(\Gamma, x) + h^\circ(\Gamma, x)
= h(\Gamma, x) + x^n h(\Gamma, 1/x) =
h_\aA(\sigma_n, x) + x^n h_\aA(\sigma_n, 1/x)$.
\end{remark}

\begin{remark} \label{rem:bar{p}(x)} \rm
The polynomial

\[ \bar{p}_\aA(\sigma_n, x) \ := \ \sum_{k=0}^n
   \bar{p}_\aA(n,k) x^k \ = \ \sum_{k=0}^n c(n,k)
	 x^k, \]

where, as in the proof of \Cref{prop:h-sdA(simplex)},
$c(n,k)$ is the number of permutations in
$\fS_n$ with excedance set equal to $[k]$, was
shown to be symmetric and unimodal in
\cite[Section~3]{ES00}. For the first few values
of $n$,

\[  \bar{p}_\aA(\sigma_n, x) \ = \ \begin{cases}
		1, & \text{if $n = 1$} \\
    1 + x, & \text{if $n=2$} \\
    1 + 3x + x^2, & \text{if $n=3$} \\
    1 + 7x + 7x^2 + x^3, & \text{if $n=4$} \\
    1 + 15x + 31x^2 + 15x^3 + x^4, &
                            \text{if $n=5$} \\
    1 + 31x + 115x^2 + 115x^3 + 31x^4 + x^5,
                           & \text{if $n=6$} \\
		1 + 63x + 391x^2 + 675x^3 + 391x^4 + 63x^5 +
		            x^6, & \text{if $n=7$.}
    \end{cases}  \]

The following statement is stronger than the
real-rootedness of $h_\aA(\sigma_n, x)$.

\begin{conjecture} \label{conj:bar{p}(x)-roots}
The polynomial $\bar{p}_\aA(\sigma_n, x)$ is
real-rooted and interlaces $\bar{p}_\aA(\sigma_{n+1},
x)$ for every $n \in \NN$. In particular,
$\bar{p}_\aA(\sigma_n, x)$ is $\gamma$-positive
for every $n \in \NN$.
\end{conjecture}

\end{remark}

\subsection{The local $h$-polynomial}
\label{sec:local}

We now focus on the local $h$-polynomial
$\ell_\aA(\sigma_n, x)$ of the antiprism
triangulation of $\sigma_n$. For the
first few values of $n$,

\[  \ell_\aA(\sigma_n, x) \ = \ \begin{cases}
    1, & \text{if $n = 0$} \\
		0, & \text{if $n = 1$} \\
    2x, & \text{if $n=2$} \\
    3x + 3x^2, & \text{if $n=3$} \\
    4x + 30x^2 + 4x^3, & \text{if $n=4$} \\
    5x + 130x^2 + 130x^3 + 5x^4, &
                            \text{if $n=5$} \\
    6x + 435x^2 + 1460x^3 + 435x^4 + 6x^5,
                           & \text{if $n=6$} \\
		7x + 1281x^2 + 10535x^3 + 10535x^4 + 1281x^5
		   + 7x^6,            & \text{if $n=7$.}
    \end{cases}  \]

We now provide a recurrence, combinatorial
interpretations and formulas for the polynomials
$\ell_\aA(\sigma_n, x)$.

\begin{proposition} \label{prop:ell-sdA}

\begin{itemize}
\itemsep=0pt
\item[{\rm (a)}]
We have

\begin{equation} \label{eq:ellA-rec}
\ell_\aA(\sigma_n, x) \ = \ \sum_{k=0}^{n-1}
{n \choose k} \ell_\aA(\sigma_k, x)
	 \left( (1+x)^{n-k} - 1 - x^{n-k} \right)
\end{equation}

for every positive integer $n$. In particular,
$\ell_\aA(\sigma_n, x)$ is unimodal for every $n
\in \NN$.

\item[{\rm (b)}]
The coefficient of $x^k$ in $\ell_\aA(\sigma_n, x)$
is equal to:

\begin{itemize}
\itemsep=0pt
\item[$\bullet$]
the number of proper multi-pointed ordered set
partitions of $[n]$ of weight $k$,

\item[$\bullet$]
the number of ways to choose an ordered set
partition $\pi$ of $[n]$ and to color $k$ elements
of $[n]$ black and the remaining $n-k$ white, so
that no block of $\pi$ is monochromatic,

\item[$\bullet$]
the number of ordered set partitions $(B_1, B_2,\ldots,B_m)$
of $[n]$ having an even number of blocks for which the union
$\bigcup_{i=1}^{m/2}B_i$ has exactly $k$ elements,

\item[$\bullet$]
${n \choose k}$ times the number of derangements
in $\fS_n$ with excedance set equal to $[k]$,

\item[$\bullet$]
the explicit expression $$ {n \choose k}
\sum_{j \ge 1} (j!)^2 S(k,j) S(n-k,j), $$
\end{itemize}

where $S(n,k)$ are the Stirling numbers of the
second kind.
\end{itemize}

\end{proposition}

\begin{proof}
The recurrence of part (a) follows from
Proposition~\ref{prop:A(uDelta)}, as a special case
of Equation~(\ref{eq:ell-A(Delta)}). The unimodality
of $\ell_\aA(\sigma_n, x)$ follows directly from the
recurrence by induction on $n$ (and, alternatively,
from the regularity of the antiprism triangulation
of the simplex; see the proof of
\Cref{prop:aASimplexToBoundary}).

For part (b), the first interpretation was already
shown in the proof of
Proposition~\ref{prop:h-sdA(simplex)} and the
second is a restatement of the first. The third interpretation
follows from the first and the proof of the corresponding
result of Proposition \ref{prop:h-sdA(simplex)} by noting that
in the provided bijection the set $[n]\setminus S$ is always empty.
Furthermore, the fifth interpretation follows from the second one
since there are ${n \choose k}$ ways to choose the
$k$ black elements of $[n]$ and for every such choice
and every $j \ge 1$, there are $j! S(k,j) \cdot j!
S(n-k,j)$ ways to choose an ordered partition of
$[n]$ with $j$ blocks, none of which is
monochromatic.

Finally, we deduce the fourth interpretation from
the corresponding result of part (b) of
Proposition~\ref{prop:h-sdA(simplex)}. Let us
use the notation adopted in the proof of that
proposition, write
$\ell_\aA(\sigma_n, x) = \sum_{k=0}^n \ell_\aA(n,k)
x^k$ for $n \in \NN$ and set $$ \ell_\aA(n,k) \ = \
{n \choose k} \bar{\ell}_\aA(n,k). $$
Then, by the second interpretation, considering
the elements $1, 2,\dots,k$ colored black and
the other elements of $[n]$ colored white,
$\bar{\ell}_\aA(n,k)$ is equal to the number of
ordered set partitions of $[n]$ with no
monochromatic block. By
Proposition~\ref{prop:h-sdA(simplex)}, under the
same coloring convention, $\bar{p}_\aA(n,k)$ is
equal to the number of ways
to choose a set $[k] \subseteq S \subseteq [n]$
and an ordered set partition of $S$ with no
monochromatic block. These interpretations imply
that

\[ \bar{p}_\aA(n,k) \ = \ \sum_{i=0}^{n-k} {n-k
   \choose i} \bar{\ell}_\aA(n-i,k) \]

for all $n, k$. Denoting by $d(n,k)$ the number of
derangements in $\fS_n$ with excedance set equal to
$[k]$, it should also be clear that

\[ c(n,k) \ = \ \sum_{i=0}^{n-k} {n-k \choose i}
   d(n-i,k) \]

for all $n, k$. By
Proposition~\ref{prop:h-sdA(simplex)}, we have
$\bar{p}_\aA(n,k) = c(n,k)$ for all $n, k$. Therefore,
the two expressions for these numbers above and an
easy induction show that $\bar{\ell}_\aA(n,k) =
d(n,k)$ for all $n, k$ and the proof follows.
\end{proof}

Following notation introduced in the previous
proof, we set $\bar{\ell}_\aA(\sigma_n, x) :=
\sum_{k=0}^n \bar{\ell}_\aA(n,k) x^k$. We note that,
since the polynomial $\ell_\aA(\sigma_n, x)$ is
symmetric with center of symmetry $n/2$, so is
$\bar{\ell}_\aA(\sigma_n, x)$.

\begin{conjecture} \label{conj:ellA(x)-roots}

\begin{itemize}
\itemsep=0pt
\item[{\rm (a)}]
The polynomial $\ell_\aA(\sigma_n, x)$ is real-rooted
and interlaces $\ell_\aA(\sigma_{n+1}, x)$ for every
$n \in \NN$. In particular, $\ell_\aA(\sigma_n, x)$
is $\gamma$-positive
for every $n \in \NN$.

\item[{\rm (b)}]
The polynomial $\bar{\ell}_\aA(\sigma_n, x)$ is
real-rooted and interlaces $\bar{\ell}_\aA(\sigma_{n+1},
x)$ for every $n \in \NN$. In particular,
$\bar{\ell}_\aA(\sigma_n, x)$ is $\gamma$-positive
for every $n \in \NN$.
\end{itemize}

\end{conjecture}

A combinatorial interpretation of the polynomial
$\theta_\aA(\sigma_n, x) := h_\aA(\sigma_n, x) -
h_\aA(\partial \sigma_n, x)$, which appeared in
Conjecture~\ref{conj:thetaA}, can be deduced from
Proposition~\ref{prop:ell-sdA}.

\begin{corollary} \label{prop:theta-sdA}
The coefficient of $x^k$ in $\theta_\aA(\sigma_n, x)$
equals the number of ways to choose an ordered set
partition $\pi$ of $[n]$ and to color $k$ elements of
$[n]$ black and the remaining $n-k$ white, so that no
block of $\pi$ is monochromatic and there is a black
element which is larger than a white element in the
last block of $\pi$.
\end{corollary}

\begin{proof}
As a consequence of \cite[Lemma~4.1]{KMS19} (and as
already discussed in the proof of
Proposition~\ref{prop:A(uDelta)}), we have

\[ \theta_\aA(\sigma_n, x) \ = \ \ell_\aA(\sigma_n,
   x) \, - \, \sum_{m=0}^{n-2} {n \choose m}
   \ell_\aA(\sigma_m, x)
	 (x + x^2 + \cdots + x^{n-m-1}) \]

for every positive integer $n$. By the second
interpretation of $\ell_\aA(\sigma_n, x)$ provided
by Proposition~\ref{prop:ell-sdA} (b), the coefficient
of $x^k$ in the sum on the right-hand side is equal
to the number of ways to choose an ordered set
partition $\pi$ of $[n]$ and to color $k$ elements of
$[n]$ black and the remaining $n-k$ white, so that no
block of $\pi$ is monochromatic and every black
element in the last block of $\pi$ is smaller than
every white element of that block. Thus, the proposed
interpretation of $\theta_\aA(\sigma_n, x)$ follows
from the previous equation and
Proposition~\ref{prop:ell-sdA}.
\end{proof}

\subsection{Face-vector transformations}
\label{sec:transform}

The general results of \cite{Ath20+} on uniform
triangulations of simplicial complexes imply that
there exist nonnegative integers $q_\aA(n,k)$ and
$p_\aA(n,k,j)$ for $n, k, j \in \NN$ with $k, j
\le n$ such that

\begin{equation}
\label{eq:f-transform}
f_{j-1}(\sd_\aA(\Delta)) \ = \ \sum_{k=j}^n
       q_\aA(k,j) f_{k-1}(\Delta)
\end{equation}

and

\begin{equation}
\label{eq:h-transform}
h_j(\sd_\aA(\Delta)) \ = \ \sum_{k=0}^n p_\aA(n,k,j)
    h_k(\Delta)
\end{equation}

for every $(n-1)$-dimensional simplicial complex
$\Delta$ and every $j \in \{0, 1,\dots,n\}$.
The former equation is easy to explain; a
simple counting argument (see the proof of
\cite[Theorem~4.1]{Ath20+}) shows its validity
when $q_\aA(n,k)$ is defined as the number of
$(k-1)$-dimensional faces in the interior of the
antiprism triangulation of $\sigma_n$. This yields
the following statement.

\begin{proposition} \label{prop:f-transform}
For all integers $n \ge 1$ and $k \in \{0,
1,\dots,n\}$, $q_\aA(n,k)$ is equal to the number
of multi-pointed ordered set partitions of $[n]$ of
weight $k$. Moreover, we have the explicit formula

\[ q_\aA(n,k) \ = \ {n \choose k} \sum_{j=0}^k j!
        S(k,j) j^{n-k} \]

where, as usual, $S(k,j)$ is a Stirling number of
the second kind.
\end{proposition}

\begin{proof}
The proposed combinatorial interpretation follows
from our previous discussion and that in
Section~\ref{sec:atriang}. To verify the formula,
we note that there are ${n \choose k} \cdot j! S(k,j)$
ways to choose a $k$-element subset $S$ of $[n]$ and
an ordered partition of $S$ with $j$ blocks and, for
each such choice there are $j^{n-k}$ ways to
distribute the remaining $n-k$ elements of $[n]$ in
the blocks so as to form a multi-pointed ordered set
partition of $[n]$ with set of chosen elements equal
to $S$.
\end{proof}

Equation~(\ref{eq:h-transform}) and the nonnegativity
of the coefficients $p_\aA(n,k,j)$ which appear there
are less obvious. Various interpretations, an explicit
formula and a recurrence are given
for these numbers in \cite{Ath20+} in the general
framework of uniform triangulations. In particular,
as shown in \cite[Corollary~5.6]{Ath20+} (and
originally by the second and third author of this
paper), the recurrence

\begin{equation}
\label{eq:recurrence}
p_\aA(n,k,j) \ = \ p_\aA(n,k-1,j) +
p_\aA(n-1,k-1,j-1) - p_\aA(n-1,k-1,j)
\end{equation}

holds for all $k, j \in \{0, 1,\dots,n\}$ with
$k \ge 1$. We will also keep in mind that $p_\aA
(n,0,j) = p_\aA(n,j)$ is the coefficient of $x^j$
in $h_\aA(\sigma_n, x)$. This observation is the
special case $\Delta = \sigma_n$ of
Equation~(\ref{eq:h-transform}).

The following combinatorial interpretations of
$p_\aA(n,k,j)$ generalize some of those given for
$p_\aA(n,k)$ in
Proposition~\ref{prop:h-sdA(simplex)}.

\begin{proposition} \label{prop:h-transform}
For all integers $n \ge 1$ and $k \in
\{0, 1,\dots,n\}$, $p_\aA(n,k,j)$ is equal to:

\begin{itemize}
\itemsep=0pt
\item[$\bullet$]
the number of ways to choose a set $[k] \subseteq
S \subseteq [n]$ and an ordered set partition $\pi$
of $S$ and to color $j$ elements of $S$ black and
the remaining elements white, so that the following
condition holds: if a block $B$ of $\pi$ is
monochromatic, then

\begin{itemize}
\itemsep=0pt
\item[$\circ$] $B$ is the first block of $\pi$,
\item[$\circ$] $B \subseteq [k]$, and
\item[$\circ$] all elements of $B$ are colored black.
\end{itemize}

\item[$\bullet$]
the number of ordered set partitions $(B_1,
B_2,\dots,B_m)$ of $[n]$ for which the following
conditions hold:

\begin{itemize}
\itemsep=0pt
\item[$\circ$]
if $m$ is even, then $\bigcup_{i=1}^{\lfloor m/2
\rfloor} B_i$ has exactly $j$ elements, and
\item[$\circ$] if $m$ is odd, then the union of
$\bigcup_{i=1}^{\lfloor m/2 \rfloor} B_i$ and $B_m
\cap [k]$ has exactly $j$ elements.
\end{itemize}

\end{itemize}

\end{proposition}

\begin{proof}
Let $Q(n,k,j)$ be the collection of triples of sets
$S$, partitions of $S$ and colorings of the elements
of $S$ described in the first proposed combinatorial
interpretation of $p_\aA(n,k,j)$ and let $q(n,k,j)$
be the cardinality of $Q(n,k,j)$.
We will show that $p_\aA(n,k,j) = q(n,k,j)$. This
is true for $k=0$ by the first combinatorial
interpretation of $p_\aA(n,j) = p_\aA(n,0,j)$
provided by Proposition~\ref{prop:h-sdA(simplex)}.
Thus, it suffices to show that the numbers $q(n,k,j)$
satisfy recurrence (\ref{eq:recurrence}) or,
equivalently, that

\[ q(n,k-1,j) \ = \ q(n,k,j) + q(n-1,k-1,j)  -
   q(n-1,k-1,j-1) \]

for $k \ge 1$. By definition, $q(n,k-1,j)$ is the
number of triples in the collection $Q(n,k-1,j)$,
each one consisting of a set $S$, a partition of $S$
and a coloring of the elements of $S$ having certain
properties. Clearly, we have $k \not\in S$ for
exactly $q(n-1,k-1,j)$ of these triples. Moreover,
we have $k \in S$ for exactly $q(n,k,j) -
q(n-1,k-1,j-1)$ of them, since for exactly
$q(n-1,k-1,j-1)$ of the triples in $Q(n,k,j)$
there is a monochromatic block which contains $k$.
This proves the first interpretation.

As an alternative proof, by computing the
coefficient of $x^j$ in the right-hand side of
\cite[Equation~(12)]{Ath20+} we get the explicit
expression

\[ p_\aA(n,k,j) \ = \ \sum_{r=0}^n \sum_{i=0}^j
   {k \choose i} {n-k \choose n-r-i}
	 \ell_\aA(r,j-i). \]

The double sum on the right side is also equal to
$q(n,k,j)$ since to choose a set $S$, a partition
$\pi$ and a coloring as in the statement of the
proposition so that a monochromatic block $B$ has exactly $i$ elements,
if present, and there is a total of $r$ elements
in the remaining blocks of $\pi$, there are
${k \choose i}{n-k \choose n-r-i}$ ways to choose
the $i$ elements of $B$ and the $n-r-i$ elements of
$[n]$ not in the blocks of $\pi$ and for each such
choice, by Proposition~\ref{prop:ell-sdA}, there
are $\ell_\aA(r,j-i)$ ways to choose the blocks of
$\pi$ other than $B$.

To prove the second interpretation, it suffices to
find a bijection from $Q(n,k,j)$ to the collection
of ordered set partitions described there. Such a
bijection can be constructed as an obvious extension
of the one provided in the proof of
Proposition~\ref{prop:h-sdA(simplex)} for the
special case $k=0$. More specifically, the elements
of the monochromatic block, if present, of a colored
ordered partition in $Q(n,k,j)$ should be included
in the last block of the ordered partition produced
by the bijection; the details are left to the
interested reader.
\end{proof}

\section{Lefschetz properties}
\label{sec:Lefschetz}

This section reviews basic definitions and background
on Lefschetz properties for simplicial complexes and
includes some preliminary technical results, which
will be applied in the following section in the context
of antiprism triangulations.

Let $\Delta$ be an $(n-1)$-dimensional simplicial
complex which is Cohen--Macaulay over an infinite
field $\FF$ and let $s \le n$ be a positive
integer. We say that $\Delta$ has the
\emph{$s$-Lefschetz property} (over $\FF$) if
there exists a linear system of parameters $\Theta$
for $\FF[\Delta]$ and a linear form $\omega \in
\FF[\Delta]$, such that the multiplication maps
$$
\cdot\omega^{s-2i}:\left(\FF[\Delta]/
\Theta\FF[\Delta]\right)_i\to\left(\FF[\Delta]/
\Theta\FF[\Delta]\right)_{s-i}
$$
are injective for all $0 \le i \le \lfloor (s-1)/2
\rfloor$. Following \cite{KN09}, we call $\Delta$
\emph{almost strong Lefschetz} (over $\FF$) if it
has the $(n-1)$-Lefschetz property. Usually, if
$\Delta$ has the $n$-Lefschetz property
and, additionally, the above multiplication maps
are isomorphisms, one says that $\Delta$ is
\emph{strong Lefschetz} (or $\Delta$ has the
\emph{strong Lefschetz property}). Lefschetz
properties are an important tool in the area of
face enumeration of simplicial complexes; various
classes of simplicial complexes, the most
prominent probably being boundary complexes of
simplicial polytopes \cite{Sta80}, are known to
have such properties. Barycentric subdivisions
of shellable simplicial complexes were shown
in~\cite{KN09} to be almost strong Lefschetz
over $\FF$.

Lefschetz properties are known to be preserved
under certain constructions. In particular, they
behave well under sufficiently nice edge
subdivisions. Before we can make this more
precise, we need to introduce some definitions.
Let $\Delta$ be a simplicial complex on a vertex
set $V$ which is endowed with a total order $<$.
Given an edge $e = \{a, b\}\in \Delta$ with $a < b$,
the \emph{contraction} $\cC_\Delta(e)$ of $\Delta$
with respect to $e$ is the simplicial complex on
the vertex set $V \sm \{b\}$ which is obtained
from $\Delta$ by identifying vertices $a$ and $b$,
i.e.,

\begin{equation*}
\cC_\Delta(e) \ := \ \{F \in \Delta~:~b \notin F\}
\, \cup \, \{(F \sm \{b\})\cup\{a\}~:~b \in F \in
\Delta\}.
\end{equation*}

We say that $\Delta$ satisfies the \emph{Link
Condition} with respect to $e$ if

\begin{equation*}
\link_\Delta(e) \ = \ \link_\Delta(\{a\}) \cap
\link_\Delta(\{b\}).
\end{equation*}

\begin{proposition} \label{prop:LinkCondition}
Let $\FF$ be an infinite field and let $\Delta$
be an $(n-1)$-dimensional Cohen--Macaulay complex
over $\FF$. Suppose $\Delta$ satisfies the
Link Condition with respect to an edge $e \in
\Delta$. If $\cC_\Delta(e)$ is Cohen--Macaulay
over $\FF$ of dimension $n-1$ and both
$\link_\Delta(e)$ and $\cC_\Delta(e)$ are strong
(respectively, almost strong) Lefschetz over
$\FF$, then so is $\Delta$.
\end{proposition}

We note that since, e.g., by Reisner's criterion
\cite{Rei76}, the Cohen--Macaulay property is
inherited by links, it is guaranteed that
$\link_{\Delta}(e)$ is Cohen--Macaulay. On the
contrary, the contraction of an edge does not even
need to be pure. \Cref{prop:LinkCondition} was
proved in \cite[Proposition 3.2]{Mur10} for the
strong Lefschetz property if $\FF$ is an arbitrary
infinite field of any characteristic (see also
\cite[Theorem 2.2]{BN10} for the same result in
characteristic zero). Since it is not entirely
obvious, although reasonable to believe, that
the proofs go through for the almost strong
Lefschetz property, we sketch the main steps of
the proof.

\medskip
\noindent
\emph{Proof of Proposition~\ref{prop:LinkCondition}}.
Let $V$ be the vertex set of $\Delta$ and $e = \{a,
b\} \in \Delta$, where $a < b$. Following
\cite{Mur10}, we consider the shift operator

\begin{align*}
C_e(F) \ = \ \begin{cases}
(F \sm \{b\}) \cup \{a\}, &\mbox{ if } b \in F, \,
a \notin F \mbox{ and } (F \sm \{b\}) \cup \{a\}
\notin \Delta,\\
F, &\mbox{ otherwise,}
\end{cases}
\end{align*}

which goes back to \cite{EKR61}, and set $\Shift_e
(\Delta) = \{C_e(F): F \in \Delta\}$. Since the Link
Condition holds for $e$, \cite[Lemma 2.1]{Mur10}
implies that
$$
\Shift_e(\Delta) \ = \ \cC_\Delta(e) \cup \{\{b\}
\cup F~:~F \in a \ast \link_\Delta(e)\}.
$$
This implies that $\Shift_e(\Delta) = \cC_\Delta(e)
\cup \Star_\Delta(e)$ and, as a result, there is the
exact sequence of $\FF[x_v: v \in V]$-modules

\begin{equation}\label{eq:exact}
0 \to \FF[\Star_\Delta(e)] \to \FF[\Shift_e(\Delta)]
  \to \FF[\cC_\Delta(e)] \to 0,
\end{equation}
where the first map is given by multiplication with
$x_b$. Since $\link_\Delta(e)$ is $(n-3)$-Lefschetz,
so is $\Star_\Delta(e)$ (see, e.g.,
\cite[Lemma 2.1]{KN09}). Hence, there exist $\Theta
= (\theta_1, \theta_2,\dots,\theta_n)$ and a linear
form $\omega \in \FF[x_v: v \in V]$ such that $\Theta$
is an l.s.o.p. for $\FF[\Star_\Delta(e)]$,
$\FF[\Shift_e(\Delta)]$ and $\FF[\cC_\Delta(e)]$
simultaneously and $\omega$ is an $(n-1)$- and
$(n-3)$-Lefschetz element for $\cC_\Delta(e)$ and
$\Star_\Delta(e)$, respectively, with respect to
$\Theta$. Hence, from \eqref{eq:exact} we get the
commutative diagram
$$\begin{array}{ccccccccc}
   0&\to& \FF[\Star_\Delta(e)]_{\ell-1} &\to& \FF[\Shift_e(\Delta)]_\ell &\to & \FF[\cC_\Delta(e)]_\ell&\to& 0\\
    & & & & & & &&\\
   & & \downarrow \omega^{n-3-2(\ell-1)}& &\ \ \downarrow \omega^{n-2\ell-1}& & \ \downarrow \omega^{n-2\ell-1}& & \\
     & & & & & & &&\\
      &\empty &  \FF[\Star_\Delta(e)]_{n-2-\ell} &\to& \FF[\Shift_e(\Delta)]_{n-1-\ell}&\to & \FF[\cC_\Delta(e)]_{n-1-\ell}&\to &0
   \end{array}$$
for $0 \leq \ell \leq \lfloor (n-1)/2\rfloor$, where
we have set $\FF[\Star_\Delta(e)]_{-1} = 0$.

Since the left and right vertical maps are injective
by assumption, so is the middle map by the snake
lemma. Thus, $\Shift_e(\Delta)$ has the almost strong
Lefschetz property. Moreover, since $\Delta$
satisfies the Link Condition with respect to $e$, we
conclude from \cite[Lemma 2.2]{Mur10} that
$I_{\Shift_e(\Delta)}$ is an initial ideal of
$I_\Delta$ with respect to a certain term order.
Finally, $\Delta$ has the $(n-1)$-Lefschetz
property by \cite[Proposition 2.9]{Wie04}. Wiebe's
orginal result was for $\mathfrak{m}$-primary
homogeneous ideals having the strong Lefschetz
property. However, the same proof works in our
setting.
\qed\\

Given a simplicial complex $\Delta$ and
face $U = \{u_1, u_2,\dots,u_n\} \in \Delta$, we
say that $\Delta$ satisfies the \emph{strong Link
Condition} with respect to $U$ if

\begin{equation}\label{eq:StrongLinkCondition}
\link_\Delta(F)\cap\link_\Delta(G) \ = \
\link_\Delta(F\cup G)
\end{equation}

for all $F,G\subseteq U$ with $F \cap G = \varnothing$.
Note that, in this case, \eqref{eq:StrongLinkCondition}
holds for all (not necessarily disjoint) subsets $F,
G\subseteq U$. The following technical lemma relates
the strong Link Condition to the usual Link Condition.

\begin{lemma}\label{lem:LinkCondition}
Let $\Delta$ be a simplicial complex which satisfies
the strong Link Condition with respect to the face
$U = \{u_1, u_2,\dots,u_n\}$. Then, $\cC_\Delta
(\{u_{n-1},u_n\})$ satisfies the strong Link Condition
with respect to $U \sm \{u_n\}$.

In particular, all edges of $2^U$ can be contracted
successively so that at each step, the Link Condition
is satisfied with respect to the contracted edge.
\end{lemma}

\begin{proof}
To simplify notation, we set $\Delta' = \cC_\Delta
(\{u_{n-1},u_n\})$ and let $U' = U \sm \{u_n\}$ be
the vertex set of $\Delta'$. We
consider disjoint sets $F, G \subseteq U'$ and
observe that, by definition of $\Delta'$,

\begin{eqnarray}
\link_{\Delta'}(F) & = &
\{H \in \link_\Delta(F)~:~u_n\notin H\}
\label{eq:link1} \\
& & \cup \ \{(H \sm \{u_n\}) \, \cup \,
\{u_{n-1}\}~:~u_n\in H\in\link_\Delta(F),\ u_{n-1}
\notin H\} \nonumber
\end{eqnarray}

if $u_{n-1} \notin F$, and

\begin{equation}
\link_{\Delta'}(F) \ = \
\{H\in \link_\Delta(F)~:~u_n\notin H\} \, \cup \,
\link_{\Delta}((F \sm \{u_{n-1}\})\cup\{u_n\})
\label{eq:link2}
\end{equation}

if $u_{n-1}\in F$. The inclusion

\[ \link_{\Delta'}(F\cup G) \ \subseteq \
   \link_{\Delta'}(F) \cap \link_{\Delta'}(G) \]

holds trivially. To prove the reverse inclusion, we
consider a face $H \in \link_{\Delta'}(F) \cap
\link_{\Delta'}(G)$ and distinguish two cases.

{\sf Case 1:} $u_{n-1}\notin F\cup G$. By
Equation~\eqref{eq:link1} for $\link_{\Delta'}(F)$
and $\link_{\Delta'}(G)$, four cases can occur.
First, assume that $H \in \link_\Delta(F) \cap
\link_\Delta(G)$. Then, by the strong Link Condition,
$H\in \link_{\Delta}(F\cup G)$ and hence $H \in
\link_{\Delta'}(F\cup G)$ by \eqref{eq:link1}. Next,
suppose that $H = (H' \sm \{u_n\})
\cup \{u_{n-1}\}$ for some $H' \in \link_\Delta(F)
\cap \link_\Delta(G)$ with $u_n \in H'$, $u_{n-1}
\notin H$. Then, the strong Link Condition implies
that $H' \in \link_\Delta(F\cup G)$ and hence $H \in
\link_{\Delta'}(F\cup G)$ by \eqref{eq:link1}.
Finally, assume that $H \in \link_\Delta(F)$ and
that $H = (H' \sm \{u_n\}) \cup \{u_{n-1}\}$ for
some $H' \in \link_\Delta(G)$ with $u_n \in H'$ and
$u_{n-1} \notin H'$. Then, $F\cup H
\in \Delta$ and $G \cup (H \sm \{u_{n-1}\}) \cup
\{u_n\} \in \Delta$. From
the strong Link Condition, we conclude that $H \sm
\{u_{n-1}\} \in \link_\Delta(F\cup G\cup\{u_n\})$,
i.e., $(H \sm \{u_{n-1}\}) \cup \{u_n\} \in
\link_{\Delta}(F\cup G)$. This, together with
\eqref{eq:link1} applied to $\link_{\Delta'}
(F\cup G)$, implies again that $H \in \link_{\Delta'}
(F\cup G)$. The remaining case follows by symmetry
from the previous one.

{\sf Case 2:} $u_{n-1}\in F\cup G$. Since $F \cap G
= \varnothing$, we may assume without loss of generality
that $u_{n-1} \in F$ and $u_{n-1} \notin G$. Since
$H \in \link_{\Delta'}(F)$ and $u_{n-1} \in F$, we
must have $u_{n-1} \notin H$. From
Equation~\eqref{eq:link1}, which applies to
$\link_{\Delta'}(G)$, and the fact that $u_{n-1} \notin
H$ we conclude that $H \in \link_\Delta(G)$. Two
subcases can occur. Suppose first that $H \in
\link_\Delta(F)$. Then, the strong Link Condition
implies that $H \in \link_\Delta(F\cup G)$ and thus
$H \in \link_{\Delta'}(F\cup G)$ by
Equation~\eqref{eq:link2}, applied to
$\link_{\Delta'}(F \cup G)$. Otherwise, $H \notin
\link_\Delta(F)$ and we must have $H \in \link_{\Delta}
((F \sm \{u_{n-1}\}) \cup \{u_n\})$ by \eqref{eq:link2}.
This implies that $H \cup (F \sm \{u_{n-1}\}) \cup
\{u_n\} \in \Delta$. Since $H \cup G\in \Delta$, from
the strong Link Condition we infer that $H \cup (F \sm
\{u_{n-1}\}) \cup \{u_n\} \cup G \in \Delta$. Since
$u_{n-1} \in F$, we conclude that $H \cup F \cup G \in
\Delta'$ and so, once again, $H \in \link_{\Delta'}
(F\cup G)$. This completes the proof of the first
statement.

For the second statement we note that if $\Delta$
satisfies the strong Link Condition with respect to
$U$, then it also satisfies the Link Condition with
respect to any edge of $2^U$. Hence, the claim follows
from successive applications of the first statement.
\end{proof}

\section{Lefschetz properties of antiprism
triangulations}
\label{sec:aLefschetz}

This section aims to prove \Cref{thm:LefschetzUnimodal},
i.e., to show that the antiprism triangulation of any
shellable simplicial complex has the almost strong
Lefschetz property over $\RR$. From this we will
infer that the $h$-vector of the antiprism
triangulation of any Cohen--Macaulay simplicial
complex is unimodal and will locate its peak.

We first show that the antiprism triangulation of the
simplex $\sigma_n$ has the almost strong Lefschetz
property over $\RR$. The next lemma will be crucial.
Recall that the (strong) Link Condition was defined in
Section~\ref{sec:Lefschetz}.

\begin{lemma}\label{lem:SLCAntiprism}
Consider an $(n-1)$-dimensional simplex $2^V$ and a
triangulation $\Delta$ of its boundary complex
$\partial(2^V)$. Then, the antiprism
$\Gamma_\aA(\Delta)$ satisfies the strong Link
Condition with respect to the set of its interior
vertices.

In particular, $\sd_\aA(\sigma_n)$ satisfies the
strong Link Condition with respect to the set of
its interior vertices.
\end{lemma}

\begin{proof}
Set $V = \{v_1, v_2,\dots,v_n\}$ and let $U = \{u_1,
u_2,\dots,u_n\}$ be the set of interior vertices of
$\Gamma_\aA(\Delta)$, linearly ordered so that
$\{u_i, v_i\} \not\in \Gamma_\aA(\Delta)$ for every
$i \in [n]$.

Let $E = \{u_i : i \in I\} \subseteq U$ for some
$I \subseteq [n]$ be nonempty and let $\bar{E} =
\{u_j : j \in [n] \sm I\}$ and $\bar{F} = \{v_j :
j \in [n] \sm I\}$ be the faces of the simplices
$2^U$ and $2^V$, respectively, which are
complementary to $E$. Then, by definition of
$\Gamma_\aA(\Delta)$,

\begin{equation*}
\link_{\Gamma_\aA(\Delta)}(E) \ = \ \Delta_\aA
(\Delta_{\bar{F}}),
\end{equation*}

where the new vertices added for the $\Delta_\aA$
construction (see Remark~\ref{rem:DeltaA}) are the
elements of $\bar{E}$ and $\Delta_{\bar{F}}$ is the
restriction of $\Delta$ to the (proper) face
$\bar{F} \in 2^V$. This directly implies that
$\Gamma_\aA(\Delta)$ satisfies the strong Link
Condition with respect to $U$.
\end{proof}

Given a Cohen--Macaulay simplicial complex $\Delta$
over a field $\FF$, we say that the contraction of
an edge $e \in
\Delta$ is \emph{admissible over $\FF$} if $\Delta$
satisfies the Link Condition with respect to $e$ and
$\link_{\Delta}(e)$ is strong Lefschetz over $\FF$.
The following proposition is, essentially, a
consequence of Lemmas~\ref{lem:LinkCondition}
and~\ref{lem:SLCAntiprism}.

\begin{proposition} \label{prop:aASimplexToBoundary}
There exists a sequence of admissible edge contractions
over $\RR$ which transforms $\sd_\aA(\sigma_n)$ into
the cone over its boundary. In particular,
$\sd_\aA(\sigma_n)$ is almost strong Lefschetz over
$\RR$, if $\partial(\sd_\aA(\sigma_{n}))$ is strong
Lefschetz over $\RR$.
\end{proposition}

\begin{proof}
As before, we let $V=\{v_1,\ldots,v_n\}$ and $U =
\{u_1,\ldots,u_n\}$ be the vertices of $\sigma_n$ and
the interior vertices of $\sd_\aA(\sigma_n)$,
respectively. By \Cref{lem:SLCAntiprism}, $\sd_\aA
(\sigma_n)$ satisfies the strong Link Condition with
respect to $U$. Thus, using \Cref{lem:LinkCondition},
we can successively contract edges from $U$, each
satisfying the Link Condition, until we reach a single
vertex $u$. The resulting complex is clearly the cone
$u \ast \partial(\sd_\aA(\sigma_n))$. If we can verify
that the intermediate complexes, appearing in this
sequence of contractions, are Cohen--Macaulay over
$\RR$ and that the links of the contracted edges are
strong Lefschetz, then \Cref{prop:LinkCondition}
implies that $\sd_\aA(\sigma_n)$ is almost strong
Lefschetz, if so is the cone
$u \ast \partial(\sd_\aA(\sigma_n))$.

To prove the missing statements, we use the fact that
$\sd_\aA(\sigma_n)$ can be constructed from $\sigma_n$
by crossing operations on its faces, starting at the
facet $V$ and moving to faces of lower dimension (see
\Cref{sec:atriang}). From this it follows that the
intermediate complexes can be constructed by first
contracting the corresponding edges in the antiprism
$\Gamma_\aA(\partial(2^V))$ and then performing
crossing operations on its boundary faces. The
antiprism $\Gamma_\aA(\partial(2^V))$ is a regular
triangulation of $2^V$ and so is any subcomplex
obtained from it by the performed edge contractions.
Since, in addition, any crossing operation can be
realized by a sequence of stellar subdivisions
(see the proof of \cite[Theorem~8]{BM87}), which are
well known to preserve regularity, we conclude that
any intermediate complex in the sequence of edge
contractions from $\sd_\aA(\sigma_n)$ to the cone
$u \ast \partial(\sd_\aA(\sigma_n))$ is a regular
triangulation of $2^V$ and, in particular,
Cohen--Macaulay over $\RR$. Moreover, the regularity
of the intermediate complexes implies that the link
of any interior edge that is contracted is a
polytopal sphere and hence strong Lefschetz over
$\RR$ \cite{Sta80}. Using
\Cref{prop:LinkCondition}, we conclude that
$\sd_\aA(\sigma_n)$ is almost strong Lefschetz over
$\RR$, if so is $u \ast \partial(\sd_\aA(\sigma_n))$.
By \cite[Lemma 2.1]{KN09}, this is the case if
$\partial(\sd_\aA(\sigma_n))$ is strong Lefschetz
over $\RR$.
\end{proof}

The next statement suffices to conclude that
$\sd_\aA(\sigma_n)$ has the almost strong Lefschetz
property over $\RR$.

\begin{proposition}\label{prop:SLPSigma}
The simplicial complex $\partial(\sd_\aA(\sigma_n))$
is combinatorially isomorphic to the boundary complex
of a simplicial polytope. In particular, it is strong
Lefschetz over $\RR$.
\end{proposition}

\begin{proof}
We use again the fact that
$\partial(\sd_\aA(\sigma_n))$ can be constructed from
$\partial \sigma_n$ by a sequence of crossing operations
(see \Cref{sec:atriang}). As already mentioned, it was
shown in the proof of Theorem 8 in \cite{BM87} that
every crossing operation can be expressed as a sequence
of stellar subdivisions. Since those preserve
polytopality, the first statement follows. The second
follows from the first and \cite{Sta80}.
\end{proof}

The next result follows by combining
Propositions~\ref{prop:aASimplexToBoundary}
and~\ref{prop:SLPSigma}.

\begin{theorem}
The simplicial complex $\sd_\aA(\sigma_n)$ is almost
strong Lefschetz over $\RR$.
\end{theorem}

\begin{remark} \rm
Since the restriction of the antiprism triangulation
$\sd_\aA(\sigma_n)$ to a face $F$ of $\sigma_n$ is
the antiprism triangulation of $2^F$, we can apply
the edge contractions from the proof of
\Cref{prop:aASimplexToBoundary} to the subdivided
faces of $\partial(\sd_\aA(\sigma_n))$, ordered by
decreasing dimension. Clearly, the simplicial
complex obtained in this way is combinatorially
isomorphic to the barycentric subdivision of
$\partial\sigma_n$. Using similar arguments as in
the proof of \Cref{prop:aASimplexToBoundary}, one
can show that all edge contractions are admissible.
Indeed, let $\Delta'$ be the simplicial complex
obtained from $\partial(\sd_\aA(\sigma_n))$ after
$i$ edge contractions. Consider a face $F \in
\sigma_n$ and let $\Delta'_F$ be the restriction
of $\Delta'$ to $F$ (which is the subcomplex of
$\Delta'$ consisting of all faces with carrier
contained in $F$). If $G \in \Delta'_F$ is
a face having all vertices in the interior of
$\Delta'_F$, then it can easily be verified that

\begin{equation}\label{eq:link}
\link_{\Delta'}(G) \ = \ \link_{\Delta'_F}(G)
\ast \{\{u_{H_1},\ldots,u_{H_r}\}~:~F\subsetneq
H_1\subsetneq \cdots \subsetneq H_r\subsetneq [n]\},
\end{equation}

where $u_H$ denotes the last interior vertex in the
sequence of contractions of a face $H \in \sigma_n$.
If $G$ is the edge to be contracted in $\Delta'$,
then the previous equation, combined with the proof
of \Cref{prop:aASimplexToBoundary}, implies that
$\Delta'$ satisfies the Link Condition with respect
to $G$. We further note that the second complex on
the right-hand side of \eqref{eq:link} is isomorphic
to the barycentric subdivision of
$\link_{\partial\sigma_n}(F)$, which itself is the
barycentric subdivision of the boundary complex of
an $(n-2-|F|)$-dimensional simplex. Thus, by \cite[Proposition 2.3]{KN09} and the
proof of Proposition~\ref{prop:aASimplexToBoundary}, both simplicial
complexes on the right-hand side of \eqref{eq:link}
are strong Lefschetz over $\RR$. From this fact and
\cite[Theorem 1.2 (i)]{BN10}, it follows that
$\link_{\Delta'}(G)$ is strong Lefschetz over $\RR$.

The previous discussion shows that there exists a
sequence of admissible edge contractions transforming
$\partial(\sd_\aA(\sigma_n))$ into
$\partial(\sd(\sigma_n))$. This provides another
proof of the second statement of \Cref{prop:SLPSigma}.
Moreover, the analogous edge contraction can be applied
to the edge links. This shows that the antiprism
triangulation of $\partial \sigma_n$ is strongly edge
decomposable (see \cite[Definition 1.1]{Mur10}), since
so is the barycentric subdivision of $\partial \sigma_n$.
\qed
\end{remark}

With \Cref{prop:SLPSigma} at hand, the key
observation to complete the proof of the first
statement of \Cref{thm:LefschetzUnimodal} is that
the proof of \cite[Theorem 1.1]{KN09}, showing that
the barycentric subdivision of any shellable simplicial
complex is almost strong Lefschetz over an infinite
field (in particular, over $\RR$), works for every
uniform triangulation which fulfills this property
for simplices. For the interested reader, and to
keep this article as self-contained as possible,
we provide a sketch of the proof.

\begin{theorem} \label{thm:LefschetzShellable}
The complex $\sd_\aA(\Delta)$ is almost strong
Lefschetz over $\RR$ for every shellable simplicial
complex $\Delta$.
\end{theorem}

\begin{proof}
Let $\dim(\Delta) = n-1$, as usual. The proof
proceeds by double induction on $n$ and the number
of facets of $\Delta$. At the
base of the induction, either $\Delta$ consists
only of vertices, in which case there is nothing
to show, or $\Delta$ is a simplex, in which case
the result follows from \Cref{prop:SLPSigma}.

For the induction step we assume that $n \ge 2$,
let $V$ be the vertex set of $\Delta$ and let $A
= \RR[x_v : v\in V]$. Consider a shelling $G_1,
G_2,\dots,G_m = G$ of $\Delta$ and set
$\widetilde{\Delta}:=\langle G_1,\dots,G_{m-1}
\rangle$ and $\tau := \widetilde{\Delta}\cap 2^G$.
There is the following exact sequence of $A$-modules:

\begin{equation} \label{eq:MV-seq}
0 \rightarrow \RR[\sd_\aA(\Delta)] \rightarrow
\RR[\sd_\aA(\widetilde{\Delta})] \oplus
\RR[\sd_\aA(2^G)] \rightarrow
\RR[\sd_\aA(\tau)]\rightarrow 0.
\end{equation}

One now chooses generic linear forms $\Theta =
\theta_1, \theta_2,\dots,\theta_n$ so that $\Theta$
is an l.s.o.p. for $\RR[\sd_\aA(\Delta)]$,
$\RR[\sd_\aA(\widetilde{\Delta})]$ and
$\RR[\sd_\aA(2^G)]$ simultaneously and $\theta_1,
\theta_2,\dots,\theta_{n-1}$ is an l.s.o.p. for
$\RR[\sd_\aA(\tau)]$. Dividing out by $\Theta$ in
\eqref{eq:MV-seq} gives rise to an exact sequence

\begin{eqnarray*}
\Tor_1(\RR[\sd_\aA(\tau)],A/\Theta)\stackrel{\delta}{\rightarrow} \RR(\sd_\aA(\Delta))\rightarrow \RR(\sd_\aA(\widetilde{\Delta}))\oplus \RR(\sd_\aA(2^G))\rightarrow \RR(\sd_\aA(\tau))\rightarrow 0,
\end{eqnarray*}

where we have written $\RR(\sd_\aA(\Delta))$ for
$\RR[\sd_\aA(\Delta)]/\Theta$ and similarly for
$\RR(\sd_\aA(\widetilde{\Delta}))$,
$\RR(\sd_\aA(\tau))$ and $\RR(\sd_\aA(2^G))$. Next,
one shows
that $\Tor_1(\RR[\sd_\aA(\tau)],A/\Theta)_i=0$ for
$0 \leq i \leq \lfloor \frac{n-2}{2}\rfloor$. This
is done exactly as in the proof of
\cite[Theorem 1.1]{KN09}. Since all maps in the
previous exact sequence preserve the grading, one
gets the commutative diagram

$$\begin{array}{ccccccc}
    \RR(\sd_\aA(\Delta))_i &\to& \RR(\sd_\aA(\widetilde{\Delta}))_i\oplus \RR(\sd_\aA(2^G))_i\\
     & & & & & &\\
     \downarrow \omega^{n-2i-1}& &\ \ \downarrow (\omega^{n-2i-1},\omega^{n-2i-1})& & \\
      & & & & & &\\
       \RR(\sd_\aA(\Delta))_{n-1-i} &\to& \RR(\sd_\aA(\widetilde{\Delta}))_{n-1-i}\oplus \RR(\sd_\aA(2^G))_{n-1-i}
   \end{array}$$

where $\omega$ is a degree one element in $A$. The
induction hypothesis implies that the multiplication
map on the right-hand side is injective for a
generic $\omega$. One concludes that the
multiplication

  \begin{equation*}
	\omega^{n-2i-1}:\hspace{5pt} \RR(\sd_\aA(\Delta))_i\rightarrow \RR(\sd_\aA(\Delta))_{n-1-i}
	\end{equation*}

is injective for $0 \leq i\leq \lfloor\frac{n-2}{2}
\rfloor$ and the proof follows.
\end{proof}

To complete the proof of \Cref{thm:LefschetzUnimodal},
we need the following properties of the numbers
$p_\aA(n,k,j)$, discussed in Section~\ref{sec:transform}.

\begin{lemma}\label{lemma:propertiesTrafo}
Let $n,k,j\in \NN$ with $k,j\leq n$.
\begin{itemize}
\item[(a)] $p_\aA(n,k,j)=p_\aA(n,n-k,n-j)$,
\item[(b)] \begin{equation*}
p_\aA(n,k,0) \, \le \, p_\aA(n,k,1) \, \le \cdots \le
\, p_\aA(n,k,\lfloor n/2\rfloor)
\end{equation*}
and
\begin{equation*}
p_\aA(n,k,n) \, \le \, p_\aA(n,k,n-1) \, \le \cdots
\le \, p_\aA(n,k,\lceil n/2\rceil).
\end{equation*}
\end{itemize}
\end{lemma}

\begin{proof}
Although part (a) follows from
\cite[Proposition 4.6 (a)]{Ath20+}, we provide a
direct combinatorial proof as follows.
Let $Q(n,k,j)$ be defined as in the proof of
\Cref{prop:h-transform}, so that $p_\aA(n,k,j) =
|Q(n,k,j)|$. Also, let $\widetilde{Q}(n,n-k,n-j)$
be the set of triples defining $Q(n,n-k,n-j)$,
except that the set $S$ which is partitioned
satisfies $\{k+1,\dots,n\} \subseteq S \subseteq [n]$,
instead of $[n-k] \subseteq S \subseteq [n]$, and
that the monochromatic block, if present, must be
contained in $\{k+1,\ldots,n\}$. Since, clearly,
$|\widetilde{Q}(n,n-k,n-j)| = p_\aA(n,n-k,n-j)$,
to prove (a) it suffices to find a
bijection from $Q(n,k,j)$ to $\widetilde{Q}
(n,n-k,n-j)$. Given a triple in $Q(n,k,j)$,
consisting of an ordered partition of $[k]
\subseteq S \subseteq [n]$ and a suitable coloring of
the elements of $S$, we construct a triple in
$\widetilde{Q}(n,n-k,n-j)$ as follows. We first
switch the colors of all elements of $S$ from white
to black and vice versa. If the first block was
monochromatic, we delete it from the partition. The
block $[n] \sm S$, if nonempty, is then added to
the constructed ordered partition as its new first
block, with all its elements colored black. We leave
to the reader to verify that this process gives a well
defined map. The inverse map can be constructed by
the same procedure, applied to the triples in
$\widetilde{Q}(n,n-k,n-j)$.

For part (b), we note that the proof of
\cite[Corollary 4.4]{KN09} works, with the symmetry
of \cite[Lemma 2.5]{BW08} replaced by that of part (a).
\end{proof}

We recall that a sequence $(a_0, a_1,\dots,a_s) \in
\NN^{s+1}$ is an \emph{M-sequence} if it is the
Hilbert function of a standard graded $\FF$-algebra,
where $\FF$ is a field. Macaulay \cite{Mac27} provided
a characterization of M-sequences by means of
numerical conditions (see, e.g., \cite{StaCCA}).
\Cref{thm:LefschetzShellable} has the following
numerical consequences for the $h$-vector of the
antiprism triangulation of a Cohen--Macaulay complex.

\begin{corollary}
Let $\Delta$ be a Cohen--Macaulay simplicial complex
of dimension $n-1$ and let $g(\sd_\aA(\Delta)) =
(1, h_1(\sd_\aA(\Delta)) -
h_0(\sd_\aA(\Delta)),\dots,h_{\lfloor n/2\rfloor}
(\sd_\aA(\Delta)) - h_{\lfloor n/2\rfloor - 1}
(\sd_\aA(\Delta)))$.

\begin{itemize}
\item[(a)] $g(\sd_\aA(\Delta))$ is an M-sequence.
\item[(b)] $h(\sd_\aA(\Delta))$ is unimodal. The peak
is at position $n/2$, if $n$ is even, and at position
$(n-1)/2$ or $(n+1)/2$, if $n$ is odd.
\item[(c)] $h_i(\sd_\aA(\Delta))\leq h_{n-1-i}
(\sd_\aA(\Delta))$ for all $0 \le i \le
\lfloor(n-2)/2\rfloor$.
\end{itemize}
\end{corollary}

\begin{proof}
Parts (a) and (c) follow by standard arguments used
when one works with Lefschetz properties, see, e.g.,
\cite[Sections 3.1--3.2]{HMMNWW13}.
For part (b), the proof of \cite[Corollary 4.7]{KN09}
works, with \Cref{lemma:propertiesTrafo} replacing
\cite[Corollary 4.4 (ii)]{KN09}.
\end{proof}

We conclude this section by recording the following
properties of the Stanley-Reisner ring of the antiprism
triangulation of any simplicial complex.

\begin{proposition}
Let $\Delta$ be a simplicial complex.

\begin{itemize}
\item[(a)] $\dim (\FF[\sd_\aA(\Delta)]) =
\dim(\FF[\Delta])$.
\item[(b)] $\depth (\FF[\sd_\aA(\Delta)]) = \depth
(\FF[\Delta])$.
\item[(c)]
\begin{equation*}\reg(\FF[\sd_\aA(\Delta)]) \ = \
\begin{cases}
\dim(\Delta), \quad \quad \mbox{if }
\widetilde{H}_{\dim(\Delta)}(\Delta;\FF) = 0,\\
\dim(\Delta)+1, \quad \mbox{if }
\widetilde{H}_{\dim(\Delta)}(\Delta;\FF) \neq 0.
\end{cases}
\end{equation*}
\end{itemize}
\end{proposition}

\begin{proof}
Part (a) is clear, since $\dim\sd_\aA(\Delta) =
\dim(\Delta)$. Part (b) follows from \cite[Theorem
3.1]{Mun84}, since $\Delta$ and $\sd_\aA(\Delta)$
have homeomorphic geometric realizations. Part (c)
follows from an application of Hochster's formula
\cite[Corollary~5.12]{MS05}; one can also mimick
the detailed argument for the barycentric subdivision
given in the proof of \cite[Proposition 2.6]{KW08}.

\end{proof}

\section{Further directions}
\label{sec:direct}

This section concludes with comments, open problems
and further directions for research.

\medskip
\noindent

1. The question asking which uniform triangulations
transform $h$-polynomials with nonnegative coefficients
into polynomials with only real roots was raised in
\cite{Ath20+}. This property has been verified for
several examples, including the prototypical one
of the barycentric subdivision \cite{BW08}, and is
conjectured in this paper for the antiprism
triangulation. We believe that this property is not
uncommon among uniform triangulations.

For instance, consider any word $w = w_1 w_2 \cdots
w_d$ with $w _i \in \{a, b\}$ for every $i \in [d]$
and let $\Delta$ be a $d$-dimensional simplicial
complex. Let $\sd_w(\Delta)$ be the triangulation
of $\Delta$ defined inductively as follows. Assume
that all faces of $\Delta$ of dimension less than
$j$ have been triangulated, for some $j \in [d]$.
Then, triangulate each $j$-dimensional face $F \in
\Delta$ by the antiprism construction, if $w_j =
a$, and by the coning construction, if $w_j = b$,
over the already triangulated boundary of $F$. By
applying this process for $j = 1, 2,\dots,d$, in
this order, we get a triangulation $\sd_w(\Delta)$
of $\Delta$ which coincides with $\sd_\aA(\Delta)$,
when $w = aa \cdots a$, and with $\sd(\Delta)$,
when $w = bb \cdots b$. It seems plausible that
this triangulation has the same property for every
$w$, but it is not easy to deduce such a statement
from the results of \cite{Ath20+} and this paper.
Example~\ref{ex:binom-euler} corresponds to the
word $w = bb \cdots ba$.

\medskip
\noindent
2. The symmetric polynomials $\ell_\aA(\sigma_n, x)$
and $\bar{p}_\aA(\sigma_n, x)$ were conjectured to be
$\gamma$-positive in Section~\ref{sec:enumerate}. It
is an interesting (and possibly challenging) open
problem to find explicit combinatorial interpretations
of the corresponding $\gamma$-coefficients. Similar
remarks apply to the symmetric polynomials
$h_\aA(\partial \sigma_n, x)$ and
$\theta_\aA(\sigma_n, x) = h_\aA(\sigma_n, x) -
h_\aA(\partial \sigma_n, x)$, already shown to be
real-rooted, and hence $\gamma$-positive, in
Section~\ref{sec:enumerate}.

\medskip
\noindent
3. The local $h$-polynomial of the barycentric
subdivision of any CW-regular subdivision of the
simplex was shown to be $\gamma$-positive
in~\cite{KMS19}. Does this hold if `barycentric
subdivision' and `CW-regular subdivision' are
replaced by `antiprism triangulation' and
`CW-regular simplicial subdivision',
respectively?

\medskip
\noindent
4. The $h$-polynomial of the barycentric subdivision
of any doubly Cohen--Macaulay level simplicial complex
and the barycentric subdivision of any triangulation
of a ball were shown to have a nonnegative real-rooted
symmetric decomposition in~\cite[Section~5]{BS20}
and~\cite[Section~8]{Ath20+}, respectively. Do these
statements hold if `barycentric subdivision' is
replaced by `antiprism triangulation'?

\medskip
\noindent
5. It is natural to ask whether, in addition to
the numerical consequences of
\Cref{thm:LefschetzUnimodal}, the algebraic part
of the statement holds for the antiprism triangulation
of any Cohen--Macaulay complex as well.

\bigskip
\noindent \textbf{Acknowledgments}. Part of this
research was conducted while the first author was
in residence at the Institute Mittag-Leffler in
Djursholm, Sweden during the `Algebraic and
Enumerative Combinatorics' program in Spring 2020,
supported by the Swedish Research Council under
grant no. 2016-06596. The second and third author
were partially supported by the German Research
Council DFG GRK-1916. The third author is grateful
to Satoshi Murai for pointing out that his original
proof of \cite[Proposition 3.2]{Mur10} can be made
less technical by using the exact sequence
\eqref{eq:exact}.