\documentclass[final]{FPSAC2022}

\usepackage{mathtools}
%\usepackage{hyperref}
%\hypersetup{
%    colorlinks=true,
%    linkcolor=blue,
%    filecolor=blue,      
%    urlcolor=blue,
%    citecolor=blue,
%}
%\usepackage{amsmath}
%\usepackage{amsthm}
%\usepackage{amssymb}
\usepackage{stmaryrd}
\usepackage{xcolor}
\usepackage{tikz}
\usepackage{tikz-cd}
\usepackage[labelformat=simple]{subcaption}
%\usepackage{url}
\usepackage{bm}

\numberwithin{equation}{section}
\numberwithin{figure}{section}

\newtheorem{thmabc}{Theorem}
\newtheorem{conabc}[thmabc]{Conjecture}
\newtheorem{thm}{Theorem}[section]
\newtheorem*{thm*}{Theorem}
\newtheorem*{con*}{Conjecture}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lemma}[thm]{Lemma}
\newtheorem{conj}[thm]{Conjecture}
\newtheorem{quest}[thm]{Question}
\newtheorem{obs}[thm]{Observation}

\renewcommand*{\thethmabc}{\Alph{thmabc}}
\renewcommand*{\theconabc}{\Alph{conabc}}

\theoremstyle{definition}
\newtheorem{defn}[thm]{Definition}
\theoremstyle{remark}
\newtheorem{remark}[thm]{Remark}
\newtheorem{ex}[thm]{Example}

\DeclareMathOperator{\Trel}{T}
\DeclareMathOperator{\Hilb}{Hilb}
\DeclareMathOperator{\sd}{sd}
\DeclareMathOperator{\des}{des}
\DeclareMathOperator{\rk}{rk}
\DeclareMathOperator{\End}{End}
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\Aut}{Aut}
\DeclareMathOperator{\GL}{GL}
\DeclareMathOperator{\PGL}{PGL}
\DeclareMathOperator{\diag}{diag}
\DeclareMathOperator{\Gal}{Gal}
\DeclareMathOperator{\Stab}{Stab}
\DeclareMathOperator{\red}{red}
\DeclareMathOperator{\rank}{rk}
\DeclareMathOperator{\corank}{corank}
\DeclareMathOperator{\codim}{rk}
\DeclareMathOperator{\ess}{ess}
\DeclareMathOperator{\im}{im}
\DeclareMathOperator{\Fix}{Fix}

\newcommand{\nofield}{K}
\newcommand{\field}{\mathbb{K}}
\newcommand{\fieldSR}{\mathbb{F}}
\newcommand{\bfr}{\bm{r}}
\newcommand{\bfs}{\bm{s}}
\newcommand{\bff}{\mathbf{f}}
\newcommand{\bfT}{\bm{T}}
\newcommand{\bfX}{\bm{X}}
\newcommand{\bfZ}{\bm{Z}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\M}{\mathbb{M}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\F}{\mathbb{F}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\C}{\mathbb{C}}
\renewcommand{\phi}{\varphi}
\renewcommand{\o}{\mathfrak{o}}
\newcommand{\pbinom}[3][]{
    \genfrac{[}{]}{0pt}{}{#2}{#3}_{\ifthenelse{\isempty{#1}}{p}{#1}}
}
\renewcommand{\leq}{\leqslant}
\renewcommand{\geq}{\geqslant}
\renewcommand{\epsilon}{\varepsilon}
\newcommand{\gp}[1]{\frac{#1}{1-#1}}
\newcommand{\pb}[2][]{(\hspace{-0.9mm}( #2 )\hspace{-0.9mm})_{#1}}
\newcommand{\m}{\mathrm{Cgp}}
\newcommand{\1}{\mathbbm{1}}
\newcommand{\Des}{\mathrm{Des}_R}
\newcommand{\comp}{\mathrm{c}}
\newcommand{\norm}[1]{\left\| #1 \right\|}
\newcommand{\lri}{\mathfrak{o}}
\newcommand{\qvar}{Q}
\renewcommand{\d}{\mathrm{d}}

\allowdisplaybreaks

\newcommand{\LoF}{\mathcal{L}}
\newcommand{\propL}{\overline{\LoF}}
\newcommand{\topL}{\widetilde{\LoF}}
\newcommand{\covecs}{\mathcal{C}}
\newcommand{\topes}{\mathcal{T}}
\newcommand{\FL}{\mathcal{F}}
\newcommand{\propF}{\overline{\FL}}
\newcommand{\A}{\mathcal{A}}
\newcommand{\fHP}{\mathsf{fHP}}
\newcommand{\cfHP}{\mathsf{cfHP}}
\newcommand{\EP}[2]{E_{#1}^{\mathsf{#2}}}
\newcommand{\zero}{\mathrm{z}}

\renewcommand\thesubfigure{(\Alph{subfigure})}

\title[Geometry of Flag Hilbert--Poincar\'e Series]{On the Geometry of Flag Hilbert--Poincar\'e Series for Matroids}


\author[L.~K\"uhne and J.~Maglione]{Lukas K\"uhne\thanks{\href{mailto:lukas.kuehne@math.uni-bielefeld.de}{lukas.kuehne@math.uni-bielefeld.de}}\addressmark{1}~and Joshua Maglione\thanks{\href{mailto:jmaglione@math.uni-bielefeld.de}{jmaglione@math.uni-bielefeld.de}}\addressmark{1}}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
%\author[Optional shorter names]{Longer names me\thanks{\href{mailto:hello@world.c}{hello@world.c}. Longer names me was partially supported by Grant 2017.11.14.$\partial$\;supp.}\addressmark{1}, \and you\addressmark{2}}

%% then use \addressmark to match authors to institutions here
    \address{ \addressmark{1}Fakult\"at f\"ur Mathematik,
      Universit\"at Bielefeld, D-33501 Bielefeld, Germany}

%% put the date of submission here
\received{25 November 2021}

%% leave this blank until submitting a revised version
\revised{}

%% put your English abstract here, or comment this out if you don't have one yet
%% please don't use custom commands in your abstract / resume, as these will be displayed online
%% likewise for citations -- please don't use \cite, and instead write out your citation as something like (author year)
\abstract{
    We extend the definition of coarse flag Hilbert--Poincar\'e series to
    matroids; these series arise in the context of local Igusa zeta functions
    associated to hyperplane arrangements. We study these series in the case of
    oriented matroids by applying geometric and combinatorial tools related to
    their topes. In this case, we prove that the numerators of these series are
    coefficient-wise bounded below by the Eulerian polynomial and equality holds
    if and only if all topes are simplicial. Moreover this yields a sufficient
    criterion for non-orientability of matroids of arbitrary rank. 
}

%% put your French abstract here, or comment this out if you don't have one
%\resume{\lipsum[2]}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{
    Coarse flag polynomial, Eulerian polynomials, Igusa zeta functions, oriented matroids
}

%% you can include your bibliography however you want, but using an external .bib file is STRONGLY RECOMMENDED and will make the editor's life much easier
%% regardless of how you do it, please use numerical citations, ie. [xx, yy] in the text

%% this sample uses biblatex, which (among other things) takes care of URLs in a more flexible way than bibtex
%% but you can use bibtex if you want
\usepackage[backend=bibtex]{biblatex}
\addbibresource{Simplicial.bib}
%% note the \printbibliography command at the end of the file which goes with these biblatex commands

\begin{document}

\maketitle
%% note that you DO NOT have to put your abstract here -- it is generated by \maketitle and the \abstract and \resume commands above



\section{Introduction}

We present the main results from~\cite{KM:simplicial}, where details and omitted
proofs can be found. The flag Hilbert--Poincar\'e series associated to a
hyperplane arrangement, defined in~\cite{MV:fHPseries}, is a rational function
in several variables connected to local Igusa zeta
functions~\cite{BSY:hyperplane-zf}. In fact, polynomial substitutions of the
variables of the flag Hilbert--Poincar\'e series also yield motivic zeta
functions associated to matroids~\cite{JKU:motivic}; see~\cite{vanderVeer} for
the topological analog. There are also substitutions yielding so-called ask zeta
functions associated to certain modules of matrices~\cite{RV:CICO};
see~\cite[Proposition~4.8]{MV:fHPseries}. 

Here we consider a specialization in variables $Y$ and $T$, called the coarse
flag Hilbert--Poincar\'e series, which seems to have remarkable combinatorial
properties. In~\cite{MV:fHPseries}, it was shown that for most Coxeter
hyperplane arrangements, the numerator of this specialization at $Y=1$ is equal
to an Eulerian polynomial. We generalize this to the setting of oriented matroids, a
combinatorial abstraction of the face structure determined by real hyperplane
arrangements. We show that the numerator can be better understood from the
geometry of the topes, which are analogs of the chambers for real hyperplane
arrangements.  This settles a question by Voll and the second
author~\cite[Question~1.7]{MV:fHPseries} for the case of real arrangements, asking
about which properties of a hyperplane arrangement guarantee the equality to
Eulerian polynomials mentioned above.


\subsection{Flag Hilbert--Poincar\'e series for matroids}

Let $M$ be a matroid, with ground set $E$, and $\LoF(M)$ its lattice of flats,
with bottom and top elements denoted by $\widehat{0}$ and $\widehat{1}$, respectively.
Relevant definitions concerning matroids and oriented matroids are given
in~\cite[Section~2]{KM:simplicial}.
We call a matroid $M$ \emph{orientable} if there exists an oriented matroid
whose underlying matroid is $M$. An orientable matroid $M$ is \emph{simplicial}
if $M$ has an oriented matroid structure such that the face lattice of every
tope is a Boolean lattice --- equivalently, for real hyperplane arrangements every
chamber is a simplicial cone. For example, all Coxeter arrangements are
simplicial.

Let $\mu_M \colon \LoF(M) \rightarrow \Z$ be the
M\"obius function on~$\LoF(M)$, where $\mu_M(\widehat{0})=1$ and
$\mu_M(X)=-\sum_{X'<X}\mu_M(X')$. A well-studied invariant of a matroid $M$ is
the \emph{Poincar\'e polynomial}  
\begin{align*}
    \pi_M(Y) &= \sum_{X\in \LoF(M)} \mu_M(X)(-Y)^{r(X)} ,
\end{align*}
where $r(X)$ is the \emph{rank} of $X$ in $\mathcal{L}(M)$, viz.~one less than
the maximum over the number of elements of all flags from $\widehat{0}$ to $X$. If
$M$ is realized by a hyperplane arrangement $\mathcal{A}$, then its Poincar\'e
polynomial captures topological and algebraic properties of
$\mathcal{A}$~\cite{OT:hyperplanes}.

For a poset $P$ let $\Delta(P)$ be the set of flags of $P$, and let
$\Delta_k(P)\subseteq \Delta(P)$ be the set of flags of size $k$.
If $P$ has a bottom element $\widehat{0}$ and a top element $\widehat{1}$ set
$\overline{P} = P\setminus\{\widehat{0},\widehat{1}\}$.
The \emph{flag Poincar\'e polynomial} associated to $F =
(X_1 < \cdots < X_{\ell}) \in \Delta(\propL(M))$, with $\ell\geq 0$, is the
product of Poincar\'e polynomials on the minors determined by $F$, 
\begin{align*} 
    \pi_F(Y) &= \prod_{k=0}^{\ell} \pi_{M/X_{k}|X_{k+1}}(Y),
\end{align*}
where $X_0=\widehat{0}$ and $X_{\ell+1}=\widehat{1}$. Here, $M/X_k$ is the contraction
of $X_k\subseteq E$ from $M$, and $M|X_{k+1}$ is the restriction of $M$ to
$X_{k+1}\subseteq E$. The lattice $\LoF(M/X_{k}|X_{k+1})$ is isomorphic to the
interval $[X_{k}, X_{k+1}]$ in $\LoF(M)$.

\begin{defn}
The \emph{coarse flag Hilbert--Poincar\'e series} of a matroid~$M$ is
\begin{align*} 
\cfHP_M(Y, T) &= \dfrac{1}{1-T}\sum_{F\in\Delta(\propL(M))} \pi_F(Y) \left(\dfrac{T}{1-T}\right)^{|F|}  = \dfrac{\mathcal{N}_M(Y, T)}{(1 - T)^{r(M)}} .
\end{align*}
We call $\mathcal{N}_M(Y, T)$ the \emph{coarse flag polynomial}:
\begin{align*}
	\mathcal{N}_M(Y, T) = \sum_{F\in\Delta(\propL(M))}  \pi_F(Y) T^{|F|}(1-T)^{r(M)-1-|F|}.
\end{align*}
\end{defn}

\subsection{Main results}

For rational polynomials $f(T)=\sum_{k\geq 0} a_kT^k$ and $g(T) = \sum_{k\geq 0}
b_kT^k$, we write $f(T)\leq g(T)$ if $a_k\leq b_k$ for all $k\geq 0$. We write
$f(T)<g(T)$ to mean $f(T)\leq g(T)$ and $f(T)\neq g(T)$. 

The \emph{Eulerian polynomials} $\EP{r+1}{A}(T)$ and $\EP{r+1}{B}(T)$ are equal
to the $h$-polynomials of the barycentric subdivisions of the boundaries of the
$r$-dimensional simplex and the cross-polytope,
respectively~\cite[Theorem~11.3]{Petersen:EulerianNumbers}. The Eulerian
polynomials are also defined by Coxeter-theoretic descent
statistics~\cite[Section~11.4]{Petersen:EulerianNumbers}.
In~\cite[Theorem~D]{MV:fHPseries}, it was shown that for all Coxeter arrangements
$\mathcal{A}$ of rank $r$, without an $\mathsf{E}_8$-factor,
$\mathcal{N}_{\mathcal{A}}(1, T)/\pi_{\mathcal{A}}(1)=\EP{r}{A}(T)$. The next
theorem generalizes this result.

\begin{thm}\label{thm:oriented-lower-bound}
    Let $M$ be an orientable matroid of rank $r$. Then 
    \begin{align} \label{eqn:lower-bound-oriented}
        E_r^{\mathsf{A}}(T) \leq \dfrac{\mathcal{N}_M(1, T)}{\pi_M(1)},
    \end{align}
    and equality holds if and only if $M$ is simplicial. Moreover, 
    \begin{align*} 
        \mathcal{N}_M(1, T^{-1}) &= T^{r-1} \mathcal{N}_M(1, T). 
    \end{align*} 
\end{thm}

The key insight in the proof for Theorem~\ref{thm:oriented-lower-bound} is that
in the orientable case $\mathcal{N}_M(1, T)$ is a sum of $h$-polynomials. Each
of the summands is determined by the topes of $M$; see
Proposition~\ref{prop:h-vectors}.
A byproduct of Theorem~\ref{thm:oriented-lower-bound} is a sufficient condition
for non-orientability of matroids. The rank $3$ case yields an inequality
concerning the the number of rank $2$ flats above every element in $M$.

\begin{cor}\label{cor:non-orientability} 
	Assume $M$ is a simple matroid with rank $3$, and suppose $c$ is the number
	of rank $2$ flats of $M$ and $s$ the sum of their sizes. If
	$3(c-1) < s$, then $M$ is non-orientable. 
\end{cor}

It is known that the Fano matroid is non-orientable, which is also shown by
Corollary~\ref{cor:non-orientability} since it has $7$ lines, each with $3$
points. There are a number of sufficient conditions for the non-orientability of
matroids. Based on experiments using the database of non-orientable
matroids~\cite{matroiddatabase}, we report that the condition in
Corollary~\ref{cor:non-orientability} is independent from the sufficient
condition in~\cite{CsimaSawyer:6n13} for rank $3$ matroids; see also
\cite[Proposition~6.6.1(i)]{OrientedMatroids}.
Moreover, Corollary~\ref{cor:non-orientability} is related to Corollary 2.6 in~\cite{CuntzGeis}
where Cuntz and Geis proved that a rank $3$ arrangement is simplicial if and only if
its underlying matroid satisfies $3(c-1)=s$ in the notation above.

The lower bound in~\eqref{eqn:lower-bound-oriented} raises the following
question. How large or how small can the coefficients of the numerator of
$\cfHP_M(1, T)/\pi_M(1)$ be? All of our results and computations suggest the
following.

\begin{conj}\label{conj:bounds}
    For all matroids $M$ of rank $r\geq 3$, 
    \begin{align*} 
        (1 + T)^{r-1} < \dfrac{\mathcal{N}_M(1, T)}{\pi_M(1)} < E_r^{\mathsf{B}}(T). 
    \end{align*}
\end{conj}

We note that $E_1^{\mathsf{A}}(T) = E_1^{\mathsf{B}}(T) = 1$ and
$E_2^{\mathsf{A}}(T) = E_2^{\mathsf{B}}(T)=1 + T$, and all matroids of rank 1 or
2 are both orientable and simplicial. For orientable matroids, the lower bound of
Conjecture~\ref{conj:bounds} holds by Theorem~\ref{thm:oriented-lower-bound}.
Moreover, the upper bound in Conjecture~\ref{conj:bounds} is reminiscent of
similar ``$f$-vector'' bounds proved in~\cite{Fukudaetal,Varchenko}.

\begin{thm}\label{thm:bounds}\mbox{}
    \begin{enumerate}[label={(\arabic*)}]
        \item \label{sharpness} If Conjecture~\ref{conj:bounds} holds, then the bounds are sharp.
        \item Conjecture~\ref{conj:bounds} holds for all matroids of rank $3$.
        Moreover for all orientable matroids, the upper bound holds for the
        linear term of the polynomials, so Conjecture~\ref{conj:bounds} holds
        for all orientable matroids of rank $4$. 
    \end{enumerate}
\end{thm}

\begin{remark} 
    To prove Theorem~\ref{thm:bounds}~\ref{sharpness} we consider two extremal
    families of matroids for the lower and upper bounds: these are the
    projective geometries and uniform matroids, respectively. See~\cite[Section~5]{KM:simplicial} for details.
\end{remark} 

In fact, more is known to hold for $\mathcal{N}_M(Y, T)$ in the case where
$r(M)\leq 3$. We prove, in Proposition~\ref{prop:rank-3}, that the numerator is
nonnegative, palindromic, and when $Y=1$ real-rooted. In particular,
Conjecture~E from~\cite{MV:fHPseries} holds for all central hyperplane arrangements with
rank at most $3$. We are also interested in whether or not these three
properties hold for the numerator of $\mathcal{N}_M(Y, T)$ for all matroids of
rank larger than $3$. For oriented matroids of rank $4$, the polynomial
$\mathcal{N}_M(1, T)$ is real-rooted, which follows from
Theorem~\ref{thm:oriented-lower-bound}. This raises the following general
question.

\begin{quest}\label{quest:realrooted}
    Is the polynomial $\mathcal{N}_M(1, T)$ real-rooted for all matroids $M$?
\end{quest}

Brenti and Welker asked whether the $h$-polynomial of the barycentric
subdivision of a general polytope is real-rooted~\cite{BrentiWelker}. In the
case of real hyperplane arrangements and their associated zonotopes, this
question is related to Question~\ref{quest:realrooted} via our geometric
interpretation of $\mathcal{N}_M(1,T)$ although the precise connection is not
yet well understood.

We give parts of the proof for Theorem~\ref{thm:oriented-lower-bound} in
Section~\ref{sec:oriented}. Section~\ref{sec:rank3} is devoted to general
matroids of rank $3$. There we also describe a pair of real hyperplane
arrangements with the same coarse flag polynomial and different underlying
matroids (Remark~\ref{rem:AB}), answering a question of Voll and the second
author~\cite{MV:fHPseries}. We conclude with some examples in Section~\ref{sec:examples}.

\section{Matroids of rank $3$}\label{sec:rank3}

We explicitly determine the coarse flag Hilbert--Poincar\'e series for
matroids of rank $3$.
First we require the next lemma, which follows from the definition of the M\"obius function. 

\begin{lem}\label{lem:poincare-poly-rank3}
    Let $M$ be a simple rank $3$ matroid on $E=[n]$. Let $c$ be the number of
    rank $2$ flats of $M$ and $s$ the sum of their sizes. Then 
    \begin{align*} 
        \pi_M(Y) &= 1 + nY + (s - c)Y^2 + (1 + s - n - c)Y^3 \\
        &= \left(1 + (n-1)Y + (1 + s - n - c)Y^2\right)(1+Y).
    \end{align*}
\end{lem}

\begin{prop}\label{prop:rank-3}
    For a simple rank $3$ matroid $M$ with ground set of size $n$, let $c$ be
    the number of rank $2$ flats of $M$ and $s$ the sum of their sizes. Then
    \begin{align*} 
       \mathcal{N}_M(Y, T) &= \pi_M(Y) + \phi_M(Y)T + Y^3\pi_M(Y^{-1})T^2 ,
    \end{align*} 
    where 
    \begin{align*} 
        \phi_M(Y) &= n+c-2 + (2s - n + c)Y + (2s - n + c)Y^2 + (n+c-2)Y^3.
    \end{align*}
\end{prop}


\begin{remark}\label{rem:AB}
    With Proposition~\ref{prop:rank-3}, we answer a question of Voll and the
    second author~\cite[Question~6.2]{MV:fHPseries}, about whether there exists a
    distinct pair of arrangements with the same coarse flag polynomial. We
    describe a pair $\mathcal{A}$ and $\mathcal{B}$ of real arrangements in
    Figure~\ref{fig:arrangements} which we found in the database
    of~\cite{Barakat} and are given by:
    \begin{align*} 
        \mathcal{A} &: xyz(x+y)(x-y)(x+2y)(x+z)(y+z)(x+y+z)=0, \\
        \mathcal{B} &: xyz(x+y)(x+2y)(x-2y)(x+z)(2y+z)(2x+2y+z)=0. 
    \end{align*}
    They both contain nine hyperplanes with $c=15$ and $s=39$
    using the above notation. The arrangement $\mathcal{A}$ has exactly two
    planes with three lines of intersection, whereas~$\mathcal{B}$
    has exactly one such plane, so they are inequivalent.
\end{remark}

\begin{figure}[h]
    \centering 
    \begin{subfigure}[b]{0.49\textwidth}
        \centering
		\includegraphics[width=.75\linewidth]{CoarseFlagExampleA}
        \caption{The arrangement $\mathcal{A}$.}
    \end{subfigure}~%
    \begin{subfigure}[b]{0.49\textwidth}
        \centering
        \includegraphics[width=.75\linewidth]{CoarseFlagExampleB}
        \caption{The arrangement $\mathcal{B}$.}
    \end{subfigure}
    \caption{Two projectivized pictures of the arrangements $\A$ and $\mathcal{B}$.}
    \label{fig:arrangements}
\end{figure}

\begin{cor} 
    If $M$ is a matroid of rank not larger than $3$, then $\mathcal{N}_M(Y, T)$
    has nonnegative coefficients and satisfies 
    \begin{align*} 
        \mathcal{N}_M(Y^{-1}, T^{-1}) &= Y^{r(M)}T^{r(M)-1}\mathcal{N}_M(Y, T).
    \end{align*}
    Moreover, the polynomial $\mathcal{N}_M(1, T)$ is real-rooted.
\end{cor}

\begin{proof} 
    This is clear if $r(M)=1$, and we assume that $M$ is a simple matroid. If
    $M$ has rank $2$, then $M\cong U_{2, n}$, where $n$ is the size of the
    ground set of $M$. Then, 
    \begin{align*} 
        \mathcal{N}_{U_{2, n}}(Y, T) &= (1+Y)(1+(n-1)Y) + (1+Y)(n-1+Y)T, 
    \end{align*}
    which satisfies the three properties. 
    
    If $r(M)=3$, then from Proposition~\ref{prop:rank-3}, $\mathcal{N}_M(Y, T)$
    satisfies 
    \begin{align*} 
        \mathcal{N}_M(Y^{-1}, T^{-1}) &= Y^{-3}T^{-2} \mathcal{N}_M(Y, T).
    \end{align*}
    By a classical result of de Bruijn and
    Erd\H{o}s~\cite{deBruijnErdos:Geometry}, the number of elements is not larger
    than the number of rank $2$ flats, so $c\geq n$. Thus, $\mathcal{N}_M(Y, T)$ has
    nonnegative coefficients. The discriminant of $\mathcal{N}_M(Y, T)$ as a
    polynomial in $T$ is
    \begin{align*} 
        \left((c+n)^2(1-Y)^2 - 4s(1 - (c+1)Y + Y^2)\right)(1 + Y)^4,
    \end{align*}
    which is positive at $Y=1$.
\end{proof}

\begin{lem}\label{lem:norm-N-rank3}
    For all matroids $M$ with rank $3$, 
    \begin{align*} 
        (1+T)^2 < \dfrac{\mathcal{N}_M(1, T)}{\pi_M(1)} < \EP{3}{B}(T) = 1 + 6T + T^2. 
    \end{align*} 
\end{lem}

\begin{proof}
    Without loss of generality, $M$ is a simple matroid. By
    Proposition~\ref{prop:rank-3}, 
    \begin{align} \label{eqn:rank3-normal}
        \dfrac{\mathcal{N}_M(1, T)}{\pi_M(1)} &= 1 + \left(2 + \dfrac{4(c-1)}{s - (c-1)}\right)T + T^2,
    \end{align} 
    where $c=|\LoF_2(M)|$ and $s=\sum_{X\in\LoF_2(M)}|X|$. Since $s\geq 2c$, 
    \begin{align*} 
        0 &< \dfrac{4(c-1)}{s - (c-1)} < 4. \qedhere
    \end{align*}
\end{proof}

We note that equation~\eqref{eqn:rank3-normal} together with
Theorem~\ref{thm:oriented-lower-bound} proves
Corollary~\ref{cor:non-orientability}.








\section{Oriented matroids}\label{sec:oriented}

Central to the proof of Theorem~\ref{thm:oriented-lower-bound} is the face
lattice of an oriented matroid $M=(E, \covecs)$.
For an oriented matroid $M$, we let $\covecs$ be the set of
covectors of $M$, $\FL(\covecs)$ the face lattice, $\topes(\covecs)$ the set of
topes, and $\zero \colon \covecs \rightarrow 2^E$ the zero map sending $C$ to $\{e\in E
~|~ C_e=0\}$.

For a poset $P$, let $\widehat{\Delta}(P)$, resp.~$\widehat{\Delta}_k(P)$, be
the set of nonempty flags, resp.~flags of length $k$, ending at a maximal
element of $P$. 

\begin{lem}\label{lem:face-poset}
    Let $M=(E,\covecs)$ be an oriented matroid of rank $r$. Then for all
    $k\in [r]$,
    \begin{align*} 
        \left| \widehat{\Delta}_k(\propF(\covecs)) \right| &= \sum_{F\in\Delta_{k-1}(\propL(M))} \pi_F(1) . 
    \end{align*}
\end{lem}


For a finite simplicial complex $\Sigma$, we write $f(\Sigma) := (f_0, \dots,
f_d)\in \N_0^{d+1}$ for the \emph{$f$-vector} of $\Sigma$, where $f_k$ is the
number of $k$-subsets in $\Sigma$---equivalently, the number of
$(k-1)$-dimensional faces. Let $f(\Sigma; T) = \sum_{k=0}^{d} f_k T^k$ be the
$f$-polynomial of $\Sigma$, and let $h(\Sigma; T) := (1 - T)^d f(\Sigma;
T/(1-T))$, which is the \emph{$h$-polynomial} associated to $\Sigma$. The
coefficients of $h(\Sigma; T)$ yield the \emph{$h$-vector} $h(\Sigma)$ of
$\Sigma$.

For a tope $\tau\in\topes(\covecs)$, we define a simplicial complex
$\Sigma(\tau) := \Delta((\widehat{0}_{\covecs}, \tau))$, which is the set of flags
in the open interval $(\widehat{0}_{\covecs}, \tau)$ in $\mathcal{F}(\covecs)$
ordered by refinement. We write $\Sigma_k(\tau)$ for the flags of $\Sigma(\tau)$
with length $k$. If $M$ is realizable over $\mathbb{R}$, then
$\Sigma(\tau)$ is the barycentric subdivision of the boundary of the chamber
determined by $\tau$. 

\begin{prop}\label{prop:h-vectors}
    Let $M=(E, \covecs)$ be an oriented matroid of rank $r$. Then 
    \begin{align*} 
       \mathcal{N}_M(1, T) &= \sum_{\tau\in\topes(\covecs)} h(\Sigma(\tau); T). 
    \end{align*}
\end{prop}

\begin{proof}
    The flags in $\widehat{\Delta}(\propF(\covecs))$ are
    partitioned into subsets $\widehat{\Delta}((\widehat{0}_{\covecs},\tau])$ for
    $\tau\in\topes(\covecs)$, and the latter are in bijection with the flags in
    $\Sigma(\tau)$. Thus,
    \begin{align*} 
        \left| \widehat{\Delta}(\propF(\covecs)) \right| &= \sum_{\tau\in\topes(\covecs)} \left| \Sigma(\tau)\right| . 
    \end{align*}
    Applying Lemma~\ref{lem:face-poset}, we have 
    \begin{align*} 
        \sum_{\tau\in\topes(\covecs)} h(\Sigma(\tau); T) &= \sum_{k=0}^{r-1}\sum_{\tau\in\topes(\covecs)} \left| \Sigma_k(\tau)\right| T^k(1-T)^{r-k-1} \\
        &= \sum_{k=0}^{r-1}~\sum_{\substack{F\in \Delta(\propL(M)) \\ |F|=k-1}} \pi_F(1) T^k(1-T)^{r-k-1} = \mathcal{N}_M(1, T).\qedhere
    \end{align*}
\end{proof}

In order to prove the lower bound in Theorem~\ref{thm:oriented-lower-bound}, we
work with the cd-index of an (Eulerian) poset. Details on the cd-index can be found
in~\cite[Ch.~3.17]{Stanley:Vol1}.

\begin{prop}\label{prop:h-lower-bound} 
    Let $M=(E,\covecs)$ be an oriented matroid of rank $r$. Then for all
    $\tau\in\topes(\covecs)$, 
    \begin{align*} 
       \EP{r}{A}(T) &\leq  h(\Sigma(\tau); T),
    \end{align*}
    and equality holds if and only if $\tau$ is a simplicial tope.
\end{prop}

\begin{proof}[Proof of Theorem~\ref{thm:oriented-lower-bound}]
    The first statement is proved using
    Propositions~\ref{prop:h-vectors} and~\ref{prop:h-lower-bound}. Since
    $\beta_{[\widehat{0}_{\covecs}, \tau]}(S) = \beta_{[\widehat{0}_{\covecs},
    \tau]}([r-1]\setminus S)$ by~\cite[Corollary~3.16.6]{Stanley:Vol1}, it
    follows that $h_k(\Sigma(\tau)) = h_{r-k-1}(\Sigma(\tau))$. Hence, the
    second statement follows by Proposition~\ref{prop:h-vectors}. 
\end{proof}


\subsection{The upper bound for the linear term}

In this section, we prove that $\mathcal{N}_M(1, T)$, for an oriented matroid
$M=(E, \covecs)$ of rank $4$, is bounded above coefficient-wise by
$\pi_M(1)\EP{r}{B}(T)$. The next lemma determines the coefficients of
$\mathcal{N}_M(1, T)$ in terms of the face lattice of $M$. To simplify notation,
we define 
\begin{align*} 
    \mathfrak{f}_{k}(\covecs) := \left|\widehat{\Delta}_{k+1}\left(\propF(\covecs)\right) \right| = \left|\left\{F\in \Delta_{k+1}\left(\propF(\covecs)\right) : F \text{ ends at a tope} \right\}\right|. 
\end{align*}
For $f(T) = \sum_{k\geq 0}a_kT^k$, let $f(T)[T^k]=a_k$.

\begin{lem}\label{lem:N-terms}
    Let $M=(E,\covecs)$ be an oriented matroid of rank $r$. For $\ell\in
    [r-1]_0$, 
    \begin{align*} 
        \mathcal{N}_M(1, T)[T^{\ell}] &= \sum_{k=0}^\ell (-1)^{\ell-k} \mathfrak{f}_{k}(\covecs) \binom{r-k-1}{\ell-k}.
    \end{align*}
\end{lem}

\begin{prop}\label{prop:rank-4}
    If $M$ is an orientable matroid of rank $r\geq 3$, then 
    \begin{align*} 
        \mathcal{N}_M(1, T)[T] < \pi_M(1) \EP{r}{B}(T)[T] . 
    \end{align*}
    If $M$ is rank $4$, then $\mathcal{N}_M(1, T)< \pi_M(1)
    E_4^{\mathsf{B}}(T)$. 
\end{prop}

\begin{proof}
    From Theorem~\ref{thm:oriented-lower-bound}, $\mathcal{N}_M(1, T)$ has
    degree $r-1$ and is palindromic. Therefore it suffices to just prove
    the inequality between the linear coefficients. 
    
    Suppose $\covecs$ is a set of covectors such that $M=(E, \covecs)$ is an
    oriented matroid. The number $\mathfrak{f}_1(\covecs)$ counts the flags of
    length two in $\propF(\covecs)$ which end at a tope, and
    $\mathfrak{f}_0(\covecs)=|\topes(\covecs)|$. Using Proposition~4.6.9
    of~\cite{OrientedMatroids}, we have the following inequality
    \begin{align}\label{eqn:Varchenko-bound}
        \mathfrak{f}_1(\covecs) &< \sum_{j=0}^{r-2} 2^{r-1-j}\binom{r-1}{j} |\topes(\covecs)|. 
    \end{align}
    Using~\cite[Section~13.1]{Petersen:EulerianNumbers}, one can express the terms
    of $\EP{r}{B}(T)$ in terms of alternating sums. The linear term is,
    thus, $3^{r-1}-r$.
    \begin{align*} 
        \mathcal{N}_M(1, T)[T] &= \mathfrak{f}_1(\covecs) - (r-1)\mathfrak{f}_0(\covecs) &  (\text{Lemma~\ref{lem:N-terms}}) & \\
        &< \left(\sum_{j=0}^{r-2} 2^{r-1-j}\binom{r-1}{j} |\topes(\covecs)|\right) - (r-1)|\topes(\covecs)| &  (\text{Equation~\ref{eqn:Varchenko-bound}}) & \\ 
        &= (3^{r-1}-1)|\topes(\covecs)| - (r-1)|\topes(\covecs)| \\
        &= \pi_M(1) \EP{r}{B}[T] .  & & 
    \end{align*}
    The penultimate equality is seen by counting, in two different ways, the
    number of ways to color $r-1$ balls with three colors. 
\end{proof}


\section{Examples}\label{sec:examples}

\subsection{A uniform matroid with rank $3$}

Consider the matroid $M=U_{3, 4}$. One set of covectors is defined by the real
arrangement given by $xyz(x+y+z)=0$. There
are $14=2^4-2$ topes since $(+++-)$ and $(---+)$ are not topes. For instance,
the inequality system given by $x>0,y>0,z>0$ and $x+y+z<0$ is infeasible. The
topes with an even number of $+$ symbols are triangles, and the topes with an
odd number of $+$ symbols are squares. Therefore, there are $8$ triangles and
$6$ squares, so by Proposition~\ref{prop:h-vectors}, 
\begin{align*} 
    \mathcal{N}_M(1, T) &= 8(1 + 4T + T^2) + 6(1 + 6T + T^2) = 14 + 68T + 14T^2.
\end{align*}
By Proposition~\ref{prop:rank-3}, the coarse numerator for $M$ is given by 
\begin{align*} 
    \mathcal{N}_M(Y, T) &= 1 + 4Y + 6Y^2 + 3Y^3 + (8 + 26Y + 26Y^2 + 8Y^3)T \\
    &\qquad + (3 + 6Y + 4Y^2 + Y^3)T^2.
\end{align*}


\subsection{A uniform matroid with rank $4$}

Three non-Coxeter, uniform matroids in~\cite{MV:fHPseries} had the seemingly
rare property that $$\mathcal{N}_M(1, T)/\pi_M(1) \in \Z[T].$$ These are the
uniform matroids $U_{r,n}$ for $(r,n)\in \{(4, 5), (4,7), (4,8)\}$, and we
consider $(r,n)=(4,7)$. From Proposition~\ref{prop:h-vectors}, this integrality
condition is equivalent to the integrality of the average of the $h$-vectors. To
do this computation, we used the hyperplane arrangement
package~\cite{KastnerPanizzut:polymake} of \textsc{polymake}~\cite{polymake}.

The matroid $U_{4, 7}$ can be realized as a hyperplane arrangement in
$\mathbb{R}^4$, whose hyperplanes are given by 
\[
x_1x_2x_3x_4(x_1+x_2+x_3+x_4)(x_1+2x_2+3x_3+4x_4)(x_1+3x_2+2x_2+5x_4)=0.
\]
There are five different polytopes corresponding to chambers of this arrangement, and they can be seen in Figure~\ref{fig:polys-U47}.
The chambers are $4$-dimensional cones over these polytopes.

\begin{figure}[h]
    \centering 
    \begin{subfigure}[b]{0.17\textwidth}
        \centering
        \begin{tikzpicture}%
            [x={(-0.979853cm, 0.048658cm)},
            y={(-0.199719cm, -0.238383cm)},
            z={(0.000070cm, 0.969952cm)},
            scale=1.000000,
            back/.style={color=black, dashed, thick},
            edge/.style={color=black, thick},
            facet/.style={fill=white,fill opacity=0.75},
            vertex/.style={inner sep=0.5pt,circle,draw=black,fill=black}]
            %
            %
            %% This TikZ-picture was produce with Sagemath version 9.2
            %% with the command: ._tikz_3d_in_3d and parameters:
            %% view = [0.0671000000000000, -0.665400000000000, -0.743500000000000]
            %% angle = 188.580000000000
            %% scale = 1
            %% edge_color = blue!95!black
            %% facet_color = blue!95!black
            %% opacity = 0.8
            %% vertex_color = green
            %% axis = False
            
            %% Coordinate of the vertices:
            %%
            \coordinate (-0.86000, -0.50000, 0.00000) at (-0.86000, -0.50000, 0.00000);
            \coordinate (0.00000, 0.00000, 1.00000) at (0.00000, 0.00000, 1.00000);
            \coordinate (0.00000, 1.00000, 0.00000) at (0.00000, 1.00000, 0.00000);
            \coordinate (0.86000, -0.50000, 0.00000) at (0.86000, -0.50000, 0.00000);
            %%
            %%
            %% Drawing edges in the back
            %%
            \draw[edge,back] (-0.86000, -0.50000, 0.00000) -- (0.86000, -0.50000, 0.00000);
            %%
            %%
            %% Drawing vertices in the back
            %%
            %%
            %%
            %% Drawing the facets
            %%
            \fill[facet] (0.00000, 1.00000, 0.00000) -- (-0.86000, -0.50000, 0.00000) -- (0.00000, 0.00000, 1.00000) -- cycle {};
            \fill[facet] (0.86000, -0.50000, 0.00000) -- (0.00000, 0.00000, 1.00000) -- (0.00000, 1.00000, 0.00000) -- cycle {};
            %%
            %%
            %% Drawing edges in the front
            %%
            \draw[edge] (-0.86000, -0.50000, 0.00000) -- (0.00000, 0.00000, 1.00000);
            \draw[edge] (-0.86000, -0.50000, 0.00000) -- (0.00000, 1.00000, 0.00000);
            \draw[edge] (0.00000, 0.00000, 1.00000) -- (0.00000, 1.00000, 0.00000);
            \draw[edge] (0.00000, 0.00000, 1.00000) -- (0.86000, -0.50000, 0.00000);
            \draw[edge] (0.00000, 1.00000, 0.00000) -- (0.86000, -0.50000, 0.00000);
            %%
            %%
            %% Drawing the vertices in the front
            %%
            \node[vertex] at (-0.86000, -0.50000, 0.00000)     {};
            \node[vertex] at (0.00000, 0.00000, 1.00000)     {};
            \node[vertex] at (0.00000, 1.00000, 0.00000)     {};
            \node[vertex] at (0.86000, -0.50000, 0.00000)     {};
            %%
            %%
        \end{tikzpicture}
        \caption{}
        \label{fig:U47-simplex}
    \end{subfigure}
    \begin{subfigure}[b]{0.17\textwidth}
        \centering 
        \begin{tikzpicture}%
            [x={(-0.957953cm, 0.076953cm)},
            y={(-0.286926cm, -0.256983cm)},
            z={(-0.000018cm, 0.963347cm)},
            scale=1.000000,
            back/.style={color=black, dashed, thick},
            edge/.style={color=black, thick},
            facet/.style={fill=white,fill opacity=0.75},
            vertex/.style={inner sep=0.5pt,circle,draw=black,fill=black}]
            %
            %
            %% This TikZ-picture was produce with Sagemath version 9.2
            %% with the command: ._tikz_3d_in_3d and parameters:
            %% view = [0.0883000000000000, -0.602500000000000, -0.793200000000000]
            %% angle = 193.260000000000
            %% scale = 1
            %% edge_color = blue!95!black
            %% facet_color = blue!95!black
            %% opacity = 0.8
            %% vertex_color = green
            %% axis = False
            
            %% Coordinate of the vertices:
            %%
            \coordinate (-0.86000, -0.50000, 0.00000) at (-0.86000, -0.50000, 0.00000);
            \coordinate (-0.86000, -0.50000, 1.00000) at (-0.86000, -0.50000, 1.00000);
            \coordinate (0.00000, 1.00000, 0.00000) at (0.00000, 1.00000, 0.00000);
            \coordinate (0.00000, 1.00000, 1.00000) at (0.00000, 1.00000, 1.00000);
            \coordinate (0.86000, -0.50000, 0.00000) at (0.86000, -0.50000, 0.00000);
            \coordinate (0.86000, -0.50000, 1.00000) at (0.86000, -0.50000, 1.00000);
            %%
            %%
            %% Drawing edges in the back
            %%
            \draw[edge,back] (-0.86000, -0.50000, 0.00000) -- (0.86000, -0.50000, 0.00000);
            %%
            %%
            %% Drawing vertices in the back
            %%
            %%
            %%
            %% Drawing the facets
            %%
            \fill[facet] (0.86000, -0.50000, 1.00000) -- (0.00000, 1.00000, 1.00000) -- (0.00000, 1.00000, 0.00000) -- (0.86000, -0.50000, 0.00000) -- cycle {};
            \fill[facet] (0.00000, 1.00000, 1.00000) -- (-0.86000, -0.50000, 1.00000) -- (-0.86000, -0.50000, 0.00000) -- (0.00000, 1.00000, 0.00000) -- cycle {};
            \fill[facet] (0.86000, -0.50000, 1.00000) -- (-0.86000, -0.50000, 1.00000) -- (0.00000, 1.00000, 1.00000) -- cycle {};
            %%
            %%
            %% Drawing edges in the front
            %%
            \draw[edge] (-0.86000, -0.50000, 0.00000) -- (-0.86000, -0.50000, 1.00000);
            \draw[edge] (-0.86000, -0.50000, 0.00000) -- (0.00000, 1.00000, 0.00000);
            \draw[edge] (-0.86000, -0.50000, 1.00000) -- (0.00000, 1.00000, 1.00000);
            \draw[edge] (-0.86000, -0.50000, 1.00000) -- (0.86000, -0.50000, 1.00000);
            \draw[edge] (0.00000, 1.00000, 0.00000) -- (0.00000, 1.00000, 1.00000);
            \draw[edge] (0.00000, 1.00000, 0.00000) -- (0.86000, -0.50000, 0.00000);
            \draw[edge] (0.00000, 1.00000, 1.00000) -- (0.86000, -0.50000, 1.00000);
            \draw[edge] (0.86000, -0.50000, 0.00000) -- (0.86000, -0.50000, 1.00000);
            %%
            %%
            %% Drawing the vertices in the front
            %%
            \node[vertex] at (-0.86000, -0.50000, 0.00000)     {};
            \node[vertex] at (-0.86000, -0.50000, 1.00000)     {};
            \node[vertex] at (0.00000, 1.00000, 0.00000)     {};
            \node[vertex] at (0.00000, 1.00000, 1.00000)     {};
            \node[vertex] at (0.86000, -0.50000, 0.00000)     {};
            \node[vertex] at (0.86000, -0.50000, 1.00000)     {};
            %%
            %%
        \end{tikzpicture}
        \caption{}
        \label{fig:U47-prism}
    \end{subfigure}
    \begin{subfigure}[b]{0.17\textwidth} 
        \centering 
        \begin{tikzpicture}%
            [x={(0.964008cm, 0.066984cm)},
            y={(-0.265872cm, 0.243004cm)},
            z={(0.000036cm, 0.967710cm)},
            scale=1.000000,
            back/.style={color=black, dashed, thick},
            edge/.style={color=black, thick},
            facet/.style={fill=white,fill opacity=0.75},
            vertex/.style={inner sep=0.5pt,circle,draw=black,fill=black}]
            %
            %
            %% This TikZ-picture was produce with Sagemath version 9.2
            %% with the command: ._tikz_3d_in_3d and parameters:
            %% view = [-0.976400000000000, 0.132200000000000, 0.171000000000000]
            %% angle = 76.7300000000000
            %% scale = 1
            %% edge_color = blue!95!black
            %% facet_color = blue!95!black
            %% opacity = 0.8
            %% vertex_color = green
            %% axis = False
            
            %% Coordinate of the vertices:
            %%
            \coordinate (1.00000, 0.00000, 0.00000) at (1.00000, 0.00000, 0.00000);
            \coordinate (0.31000, 0.95000, 0.00000) at (0.31000, 0.95000, 0.00000);
            \coordinate (-0.81000, 0.59000, 0.00000) at (-0.81000, 0.59000, 0.00000);
            \coordinate (-0.81000, -0.59000, 0.00000) at (-0.81000, -0.59000, 0.00000);
            \coordinate (0.31000, -0.95000, 0.00000) at (0.31000, -0.95000, 0.00000);
            \coordinate (1.00000, 0.00000, 1.00000) at (1.00000, 0.00000, 1.00000);
            \coordinate (0.31000, 0.95000, 0.61878) at (0.31000, 0.95000, 0.61878);
            \coordinate (0.31000, -0.95000, 0.61878) at (0.31000, -0.95000, 0.61878);
            %%
            %%
            %% Drawing edges in the back
            %%
            \draw[edge,back] (1.00000, 0.00000, 0.00000) -- (0.31000, 0.95000, 0.00000);
            \draw[edge,back] (0.31000, 0.95000, 0.00000) -- (-0.81000, 0.59000, 0.00000);
            \draw[edge,back] (0.31000, 0.95000, 0.00000) -- (0.31000, 0.95000, 0.61878);
            %%
            %%
            %% Drawing vertices in the back
            %%
            \node[vertex] at (0.31000, 0.95000, 0.00000)     {};
            %%
            %%
            %% Drawing the facets
            %%
            \fill[facet] (0.31000, -0.95000, 0.61878) -- (-0.81000, -0.59000, 0.00000) -- (0.31000, -0.95000, 0.00000) -- cycle {};
            \fill[facet] (1.00000, 0.00000, 1.00000) -- (0.31000, -0.95000, 0.61878) -- (-0.81000, -0.59000, 0.00000) -- (-0.81000, 0.59000, 0.00000) -- (0.31000, 0.95000, 0.61878) -- cycle {};
            \fill[facet] (0.31000, -0.95000, 0.61878) -- (0.31000, -0.95000, 0.00000) -- (1.00000, 0.00000, 0.00000) -- (1.00000, 0.00000, 1.00000) -- cycle {};
            %%
            %%
            %% Drawing edges in the front
            %%
            \draw[edge] (1.00000, 0.00000, 0.00000) -- (0.31000, -0.95000, 0.00000);
            \draw[edge] (1.00000, 0.00000, 0.00000) -- (1.00000, 0.00000, 1.00000);
            \draw[edge] (-0.81000, 0.59000, 0.00000) -- (-0.81000, -0.59000, 0.00000);
            \draw[edge] (-0.81000, 0.59000, 0.00000) -- (0.31000, 0.95000, 0.61878);
            \draw[edge] (-0.81000, -0.59000, 0.00000) -- (0.31000, -0.95000, 0.00000);
            \draw[edge] (-0.81000, -0.59000, 0.00000) -- (0.31000, -0.95000, 0.61878);
            \draw[edge] (0.31000, -0.95000, 0.00000) -- (0.31000, -0.95000, 0.61878);
            \draw[edge] (1.00000, 0.00000, 1.00000) -- (0.31000, 0.95000, 0.61878);
            \draw[edge] (1.00000, 0.00000, 1.00000) -- (0.31000, -0.95000, 0.61878);
            %%
            %%
            %% Drawing the vertices in the front
            %%
            \node[vertex] at (1.00000, 0.00000, 0.00000)     {};
            \node[vertex] at (-0.81000, 0.59000, 0.00000)     {};
            \node[vertex] at (-0.81000, -0.59000, 0.00000)     {};
            \node[vertex] at (0.31000, -0.95000, 0.00000)     {};
            \node[vertex] at (1.00000, 0.00000, 1.00000)     {};
            \node[vertex] at (0.31000, 0.95000, 0.61878)     {};
            \node[vertex] at (0.31000, -0.95000, 0.61878)     {};
            %%
            %%
        \end{tikzpicture}
        \caption{}
        \label{fig:U47-pinched}
    \end{subfigure} 
    \begin{subfigure}[b]{0.17\textwidth} 
        \centering 
        \begin{tikzpicture}%
            [x={(-0.723200cm, -0.312625cm)},
            y={(0.690639cm, -0.327354cm)},
            z={(-0.000008cm, 0.891687cm)}, 
            scale=1.15,
            back/.style={color=black, dashed, thick},
            edge/.style={color=black, thick},
            facet/.style={fill=white,fill opacity=0.75},
            vertex/.style={inner sep=0.5pt,circle,draw=black,fill=black}]
            %
            %
            %% This TikZ-picture was produce with Sagemath version 9.2
            %% with the command: ._tikz_3d_in_3d and parameters:
            %% view = [-0.235100000000000, -0.492400000000000, -0.838000000000000]
            %% angle = 136.380000000000
            %% scale = 1
            %% edge_color = blue!95!black
            %% facet_color = blue!95!black
            %% opacity = 0.8
            %% vertex_color = green
            %% axis = False
            
            %% Coordinate of the vertices:
            %%
            \coordinate (0.33000, 0.00000, 0.00000) at (0.33000, 0.00000, 0.00000);
            \coordinate (-0.16500, -0.28579, 0.00000) at (-0.16500, -0.28579, 0.00000);
            \coordinate (-0.16500, 0.28579, 0.00000) at (-0.16500, 0.28579, 0.00000);
            \coordinate (0.66000, 0.00000, 0.61878) at (0.66000, 0.00000, 0.61878);
            \coordinate (-0.33000, -0.57158, 0.61878) at (-0.33000, -0.57158, 0.61878);
            \coordinate (-0.33000, 0.57158, 0.61878) at (-0.33000, 0.57158, 0.61878);
            \coordinate (-0.43165, 0.00000, 1.00000) at (-0.43165, 0.00000, 1.00000);
            \coordinate (0.21583, 0.37382, 1.00000) at (0.21583, 0.37382, 1.00000);
            \coordinate (0.21583, -0.37382, 1.00000) at (0.21583, -0.37382, 1.00000);
            \coordinate (0.00000, 0.00000, 1.18524) at (0.00000, 0.00000, 1.18524);
            %%
            %%
            %% Drawing edges in the back
            %%
            \draw[edge,back] (0.33000, 0.00000, 0.00000) -- (-0.16500, -0.28579, 0.00000);
            \draw[edge,back] (-0.16500, -0.28579, 0.00000) -- (-0.16500, 0.28579, 0.00000);
            \draw[edge,back] (-0.16500, -0.28579, 0.00000) -- (-0.33000, -0.57158, 0.61878);
            \draw[edge,back] (-0.33000, -0.57158, 0.61878) -- (-0.43165, 0.00000, 1.00000);
            \draw[edge,back] (-0.33000, -0.57158, 0.61878) -- (0.21583, -0.37382, 1.00000);
            %%
            %%
            %% Drawing vertices in the back
            %%
            \node[vertex] at (-0.16500, -0.28579, 0.00000)     {};
            \node[vertex] at (-0.33000, -0.57158, 0.61878)     {};
            %%
            %%
            %% Drawing the facets
            %%
            \fill[facet] (0.00000, 0.00000, 1.18524) -- (-0.43165, 0.00000, 1.00000) -- (-0.33000, 0.57158, 0.61878) -- (0.21583, 0.37382, 1.00000) -- cycle {};
            \fill[facet] (0.00000, 0.00000, 1.18524) -- (0.21583, 0.37382, 1.00000) -- (0.66000, 0.00000, 0.61878) -- (0.21583, -0.37382, 1.00000) -- cycle {};
            \fill[facet] (0.21583, 0.37382, 1.00000) -- (0.66000, 0.00000, 0.61878) -- (0.33000, 0.00000, 0.00000) -- (-0.16500, 0.28579, 0.00000) -- (-0.33000, 0.57158, 0.61878) -- cycle {};
            %%
            %%
            %% Drawing edges in the front
            %%
            \draw[edge] (0.33000, 0.00000, 0.00000) -- (-0.16500, 0.28579, 0.00000);
            \draw[edge] (0.33000, 0.00000, 0.00000) -- (0.66000, 0.00000, 0.61878);
            \draw[edge] (-0.16500, 0.28579, 0.00000) -- (-0.33000, 0.57158, 0.61878);
            \draw[edge] (0.66000, 0.00000, 0.61878) -- (0.21583, 0.37382, 1.00000);
            \draw[edge] (0.66000, 0.00000, 0.61878) -- (0.21583, -0.37382, 1.00000);
            \draw[edge] (-0.33000, 0.57158, 0.61878) -- (-0.43165, 0.00000, 1.00000);
            \draw[edge] (-0.33000, 0.57158, 0.61878) -- (0.21583, 0.37382, 1.00000);
            \draw[edge] (-0.43165, 0.00000, 1.00000) -- (0.00000, 0.00000, 1.18524);
            \draw[edge] (0.21583, 0.37382, 1.00000) -- (0.00000, 0.00000, 1.18524);
            \draw[edge] (0.21583, -0.37382, 1.00000) -- (0.00000, 0.00000, 1.18524);
            %%
            %%
            %% Drawing the vertices in the front
            %%
            \node[vertex] at (0.33000, 0.00000, 0.00000)     {};
            \node[vertex] at (-0.16500, 0.28579, 0.00000)     {};
            \node[vertex] at (0.66000, 0.00000, 0.61878)     {};
            \node[vertex] at (-0.33000, 0.57158, 0.61878)     {};
            \node[vertex] at (-0.43165, 0.00000, 1.00000)     {};
            \node[vertex] at (0.21583, 0.37382, 1.00000)     {};
            \node[vertex] at (0.21583, -0.37382, 1.00000)     {};
            \node[vertex] at (0.00000, 0.00000, 1.18524)     {};
            %%
            %%
        \end{tikzpicture}
        \caption{}
        \label{fig:U47-beaut}
    \end{subfigure}
    \begin{subfigure}[b]{0.17\textwidth} 
        \centering 
        \begin{tikzpicture}%
            [x={(-0.623765cm, -0.245664cm)},
            y={(0.781612cm, -0.196069cm)},
            z={(0.000014cm, 0.949319cm)},
            scale=1.000000,
            back/.style={color=black, dashed, thick},
            edge/.style={color=black, thick},
            facet/.style={fill=white,fill opacity=0.75},
            vertex/.style={inner sep=0.5pt,circle,draw=black,fill=black}]
            %
            %
            %% This TikZ-picture was produce with Sagemath version 9.2
            %% with the command: ._tikz_3d_in_3d and parameters:
            %% view = [-0.250500000000000, -0.613700000000000, -0.748700000000000]
            %% angle = 146.010000000000
            %% scale = 1
            %% edge_color = blue!95!black
            %% facet_color = blue!95!black
            %% opacity = 0.8
            %% vertex_color = green
            %% axis = False
            
            %% Coordinate of the vertices:
            %%
            \coordinate (1.00000, 0.00000, 0.00000) at (1.00000, 0.00000, 0.00000);
            \coordinate (0.50000, -0.86603, 0.00000) at (0.50000, -0.86603, 0.00000);
            \coordinate (-0.50000, -0.86603, 0.00000) at (-0.50000, -0.86603, 0.00000);
            \coordinate (-1.00000, 0.00000, 0.00000) at (-1.00000, 0.00000, 0.00000);
            \coordinate (-0.50000, 0.86603, 0.00000) at (-0.50000, 0.86603, 0.00000);
            \coordinate (0.50000, 0.86603, 0.00000) at (0.50000, 0.86603, 0.00000);
            \coordinate (0.86603, 0.50000, 0.45000) at (0.86603, 0.50000, 0.45000);
            \coordinate (0.00000, -1.00000, 0.45000) at (0.00000, -1.00000, 0.45000);
            \coordinate (-0.86603, 0.50000, 0.45000) at (-0.86603, 0.50000, 0.45000);
            \coordinate (0.00000, 0.00000, 1.06471) at (0.00000, 0.00000, 1.06471);
            %%
            %%
            %% Drawing edges in the back
            %%
            \draw[edge,back] (0.50000, -0.86603, 0.00000) -- (-0.50000, -0.86603, 0.00000);
            \draw[edge,back] (-0.50000, -0.86603, 0.00000) -- (-1.00000, 0.00000, 0.00000);
            \draw[edge,back] (-0.50000, -0.86603, 0.00000) -- (0.00000, -1.00000, 0.45000);
            \draw[edge,back] (-1.00000, 0.00000, 0.00000) -- (-0.50000, 0.86603, 0.00000);
            \draw[edge,back] (-1.00000, 0.00000, 0.00000) -- (-0.86603, 0.50000, 0.45000);
            %%
            %%
            %% Drawing vertices in the back
            %%
            \node[vertex] at (-0.50000, -0.86603, 0.00000)     {};
            \node[vertex] at (-1.00000, 0.00000, 0.00000)     {};
            %%
            %%
            %% Drawing the facets
            %%
            \fill[facet] (0.00000, 0.00000, 1.06471) -- (0.86603, 0.50000, 0.45000) -- (0.50000, 0.86603, 0.00000) -- (-0.50000, 0.86603, 0.00000) -- (-0.86603, 0.50000, 0.45000) -- cycle {};
            \fill[facet] (0.86603, 0.50000, 0.45000) -- (1.00000, 0.00000, 0.00000) -- (0.50000, 0.86603, 0.00000) -- cycle {};
            \fill[facet] (0.00000, 0.00000, 1.06471) -- (0.86603, 0.50000, 0.45000) -- (1.00000, 0.00000, 0.00000) -- (0.50000, -0.86603, 0.00000) -- (0.00000, -1.00000, 0.45000) -- cycle {};
            %%
            %%
            %% Drawing edges in the front
            %%
            \draw[edge] (1.00000, 0.00000, 0.00000) -- (0.50000, -0.86603, 0.00000);
            \draw[edge] (1.00000, 0.00000, 0.00000) -- (0.50000, 0.86603, 0.00000);
            \draw[edge] (1.00000, 0.00000, 0.00000) -- (0.86603, 0.50000, 0.45000);
            \draw[edge] (0.50000, -0.86603, 0.00000) -- (0.00000, -1.00000, 0.45000);
            \draw[edge] (-0.50000, 0.86603, 0.00000) -- (0.50000, 0.86603, 0.00000);
            \draw[edge] (-0.50000, 0.86603, 0.00000) -- (-0.86603, 0.50000, 0.45000);
            \draw[edge] (0.50000, 0.86603, 0.00000) -- (0.86603, 0.50000, 0.45000);
            \draw[edge] (0.86603, 0.50000, 0.45000) -- (0.00000, 0.00000, 1.06471);
            \draw[edge] (0.00000, -1.00000, 0.45000) -- (0.00000, 0.00000, 1.06471);
            \draw[edge] (-0.86603, 0.50000, 0.45000) -- (0.00000, 0.00000, 1.06471);
            %%
            %%
            %% Drawing the vertices in the front
            %%
            \node[vertex] at (1.00000, 0.00000, 0.00000)     {};
            \node[vertex] at (0.50000, -0.86603, 0.00000)     {};
            \node[vertex] at (-0.50000, 0.86603, 0.00000)     {};
            \node[vertex] at (0.50000, 0.86603, 0.00000)     {};
            \node[vertex] at (0.86603, 0.50000, 0.45000)     {};
            \node[vertex] at (0.00000, -1.00000, 0.45000)     {};
            \node[vertex] at (-0.86603, 0.50000, 0.45000)     {};
            \node[vertex] at (0.00000, 0.00000, 1.06471)     {};
            %%
            %%
        \end{tikzpicture}
        \caption{}
        \label{fig:U47-truncated}
    \end{subfigure}
    \caption{The five different polytopes arising as chambers in the $U_{4, 7}$ arrangement.}
    \label{fig:polys-U47}
\end{figure}



There are a total of 84 chambers; 22 are simplices, 22 are triangular prisms, 30
are the polytopes seen in Figure~\ref{fig:U47-pinched}, six are the polytopes
seen in Figure~\ref{fig:U47-beaut}, and four are truncated simplices as seen in
Figure~\ref{fig:U47-truncated}. The $h$-vectors of the barycentric subdivisions
are palindromic, and the first values different from $1$ are $11$, $17$, $23$,
$29$, and $29$ respectively.

Thus, 
\begin{align*} 
    \mathcal{N}_{U_{4, 7}}(1, T) &= 22(1 + 11T + 11T^2 + T^3) + 22(1 + 17T + 17T^2 + T^3) \\
    &\qquad + 30(1 + 23T + 23T^2 + T^3) + (4+6)(1 + 29T + 29T^2 + T^3). 
\end{align*}
This has the nice coincidence that 
\begin{align*} 
    \mathcal{N}_{U_{4, 7}}(1, T) &= 84 (1 + 19T + 19T^2 + T^3).
\end{align*} 
Curiously, $(1,19,19,1)$ is the $h$-vector of the barycentric subdivision of the pyramid over a pentagon.

\acknowledgements{The authors were supported by DFG-grant VO~1248/4-1 project number~373111162.}


\printbibliography

\end{document}
