\documentclass[final]{FPSAC2022}

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\def \mcL {\mathcal{L}}
\def \Gri {\mathcal{O}}
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\def \Z {\mathbb{Z}}
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% HYPERPLANE COMMANDS
\newcommand{\mcP}{\mathcal{P}}
\newcommand{\mcN}{\mathcal{N}}
\newcommand{\fHP}{\mathsf{fHP}}
\newcommand{\cfHP}{\mathsf{cfHP}}
\newcommand{\mcF}{\mathcal{F}}
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\title[Flag Hilbert--Poincar\'e Series of
  Hyperplane Arrangements]{Flag Hilbert--Poincar\'e Series and Igusa
  Zeta Functions of Hyperplane Arrangements}

      
 \author[J.~Maglione and C.~Voll]{Joshua
   Maglione\thanks{\href{mailto:jmaglione@math.uni-bielefeld.de}{jmaglione@math.uni-bielefeld.de}. Partially supported by Deutsche Forschungsgemeinschaft grants FR~1639/4-1 and VO~1248/4-1 (project number~373111162).}\addressmark{1} \and
   Christopher Voll\thanks{\href{mailto:C.Voll.98@cantab.net}{C.Voll.98@cantab.net}}\addressmark{1}}

%% define your authors in the usual way
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%\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}}

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    \address{ \addressmark{1}Fakult\"at f\"ur Mathematik,
      Universit\"at Bielefeld, D-33501 Bielefeld, Germany}

%% put the date of submission here
\received{20 November 2019}

%% leave this blank until submitting a revised version
\revised{27 March 2020}

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\abstract{
  We introduce and study a class of multivariate rational functions
  associated with hyperplane arrangements, called flag
  Hilbert--Poincar\'e series.  These series are intimately connected
  with Igusa local zeta functions of products of linear polynomials,
  and their motivic and topological relatives. Our main results
  include a self-reciprocity result for central arrangements defined
  over fields of characteristic zero. We also prove combinatorial
  formulae for a specialization of the flag Hilbert--Poincar\'e series
  for irreducible Coxeter arrangements of types $\mathsf{A}$,
  $\mathsf{B}$, and $\mathsf{D}$ in terms of total partitions of the
  respective types. We show that a different specialization of the
  flag Hilbert--Poincar\'e series, which we call the coarse flag
  Hilbert--Poincar\'e series, exhibits intriguing nonnegativity
  features and --- in the case of Coxeter arrangements --- connections
  with Eulerian polynomials. For numerous classes and examples of
  hyperplane arrangements, we determine their (coarse) flag
  Hilbert--Poincar\'e series. Some computations were aided by a
  \SageMath~package we developed.
}

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%\resume{\lipsum[2]}

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\keywords{
  Hyperplane arrangements, Igusa's local zeta function,
  Eulerian polynomials, Stirling numbers of the second kind, Hilbert
  series, Hadamard products, topological zeta functions, total
  partitions, representable matroids, Coxeter arrangements, braid
  arrangements
}

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\begin{document}

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

We present some of the main results of~\cite{MV_IJM/21}, an abridged
version of~\cite{MV/21}.  A hyperplane arrangement
over a field $\field$ is a finite set $\A$ of affine hyperplanes in $\field^d$
for some integer~$d := \dim(\A)$. We introduce and study a multivariate rational
function $\fHP_{\A}(Y,\bfT)\in\Q(\bfT)[Y]$, called the \emph{flag
  Hilbert--Poincar\'e series} of $\A$, encompassing much of the topology and
combinatorics of~$\A$. As we shall explain, various substitutions yield
connections to several enumeration problems associated with hyperplane
arrangements.

In order to define $\fHP_{\A}$, we introduce some further
notation. Let $\intpos(\A)$ be the intersection poset of $\A$, ordered
by reverse-inclusion.  Two hyperplane arrangements are
\emph{equivalent} if their intersection posets are isomorphic. We
denote by $\widehat{0}$ (resp.\ $\widehat{1}$) the bottom (resp.\ top) element
of a poset (provided $\widehat{1}$ exists). Observe that $\widehat{1}\in
\intpos(\A)$ if and only if $\A$ is central,
\textit{i.e.}~$\bigcap_{H\in\A}H\neq\varnothing$. Define $\toplat(\A) =
\intpos(\A)\setminus\{\widehat{0}\}$ and $\proplat(\A) =
\intpos(\A)\setminus \{\widehat{0},\widehat{1}\}$. For $x\in\intpos(\A)$, we
write $\rank(x):=\rank_{\intpos(\A)}(x)$ for the \emph{rank} of~$x$,
viz.\ the supremum over the lengths of all chains from $\widehat{0}$ to
$x$.  For a poset~$P$, the \emph{order complex} $\Delta(P)$ associated
with $P$ is the simplicial complex with vertex set $P$, whose
simplices are the flags of~$P$. For $x\in\intpos(\A)$, define
hyperplane arrangements
\begin{alignat*}{2}
    \A_x &= \left\{H\in\A ~\middle|~ x\subseteq H\right\}
    &&\textup{(subarrangement),}\\ \A^x &= \left\{x\cap H ~\middle|~
    H\in\A\setminus \A_x, \; x\cap H \neq \emptyset \right\} \quad
    &&\textup{(restriction)}.
\end{alignat*}
Set $\A_{\emptyset} := \A$. Interlacing these constructions we obtain, for
$x,y\in \intpos(\A)$, the arrangement $\A^x_{y} := (\A^x)_{y} = (\A_{y})^x$.
Recall further the \emph{Poincar\'e polynomial}~\cite[Definition~2.48]{OrlikTerao/92}
\begin{equation*} 
    \pi_{\A}(Y) =\sum_{x\in \intpos(\A)} \mu(\widehat{0}, x)(-Y)^{\rk(x)} \in \Z[Y]
\end{equation*}
associated with~$\A$, where $\mu$ is the M\"obius function
on~$\intpos(\A)$; \textit{cf.}~\cite[Definition~3.15]{Stanley:Vol1}.  The Poincar\'e
polynomial is closely related to the characteristic polynomial
$\chi_{\A}(Y)$ of $\A$ via the identity (see
\cite[Definition~2.52]{OrlikTerao/92})
  \begin{equation*}
    \chi_{\A}(Y) = Y^d\pi_{\A}(-Y^{-1}).
  \end{equation*} 
We require the following
flag generalization: for $F = (x_1 < x_2 < \cdots < x_{\ell})
\in\Delta(\intpos(\A))$ (possibly empty), set $x_0=\widehat{0}$ and
$x_{\ell+1} = \emptyset$, and define
\begin{align*}
   \pi_F (Y) &= \prod_{k=0}^{\ell} \pi_{\A_{x_{k+1}}^{x_k}}(Y) \in
   \Z[Y].
\end{align*} 

The following function is the main protagonist of~\cite{MV/21}.

\begin{defn}\label{def:algebraic}
    Let $\bfT := (T_x)_{x\in \toplat{(\A)}}$ be indeterminates. The \emph{flag
    Hilbert--Poincar\'e series} associated with $\A$ is
    \begin{align*} 
        \fHP_{\A}(Y,\bfT) &= \sum_{F\in\Delta(\toplat(\A))} \pi_F(Y)
        \prod_{x\in F}\dfrac{T_x}{1 - T_x} \in\Q(\bfT)[Y].
    \end{align*}
\end{defn} 

If $\A$ is central, then
$$ \fHP_{\A}(Y,\bfT) = \frac{1}{1-T_{\widehat{1}}}
\sum_{F\in\Delta(\proplat(\A))}\pi_F(Y) \prod_{x\in F}\dfrac{T_x}{1 -
  T_x}.
$$
We remark that $\fHP_{\A}(0,\bfT)$ is the
(fine) Hilbert series of the Stanley--Reisner ring of the order
complex of $\toplat(\A)$; see Proposition~\ref{prop:coarse.Y=0}. 

\begin{ex}\label{ex:fHP-A2}
  Suppose $\mathsf{A}_2:=\{H_{ij}~|~ 1\leq i<j\leq 3\}\subset \Q^3$ is
  the arrangement, where $H_{ij}$ is the set of zeros to
  $X_i-X_j$. This is also known as the Coxeter arrangement of type
  $\mathsf{A}_2$. Then, writing $T_{ij}$ for $T_{H_{ij}}$, we have
  \begin{align*} 
    \fHP_{\mathsf{A}_2}(Y,\bfT) &= \dfrac{1}{1-T_{\widehat{1}}} \left((1+Y)(1+2Y) + (1 + Y) \left(\dfrac{T_{12}}{1-T_{12}} + \dfrac{T_{13}}{1-T_{13}} + \dfrac{T_{23}}{1-T_{23}}\right)\right) . 
  \end{align*} 
\end{ex}

The following self-reciprocity result for central arrangements over
fields of characteristic zero is our first main theorem. The
\emph{rank} of $\A$, denoted by $\rank(\A)$, is the rank of a maximal
element of $\intpos(\A)$.

\begin{thmabc}[Self-reciprocity]\label{thm:reciprocity.new}
  Let $\A$ be a central hyperplane arrangement over a field of
  characteristic zero. Then
    \begin{align*} 
        \fHP_{\A}\left(Y^{-1}, (T_x^{-1})_{x\in\toplat(\A)}\right) &=
        (-Y)^{-\rank(\A)}T_{\widehat{1}}\cdot \fHP_{\A}(Y,\bfT).
    \end{align*}
\end{thmabc}

The restriction to fields of characteristic zero reflects our method of proof rather than any known counterexamples in positive characteristic. 

\medskip

Various substitutions of the variables of the flag Hilbert--Poincar\'e series
yield connections to seemingly different enumeration problems:

First, we explain in Section~\ref{subsec:Igusa.intro} that flag
Hilbert--Poincar\'e series encode the same information as certain $\mfp$-adic
integrals associated with hyperplane arrangements (see
Theorem~\ref{thm:equivalence}). In Section~\ref{subsec:Igusa-topological} we
explicate the specific connections to the well-studied class of (both uni- and
multivariate) \emph{Igusa local zeta functions} associated with products of
linear polynomials. In~\cite[Section~2.2]{MV/21}, we also consider topological
zeta functions, which are closely related to the Igusa zeta functions. Another
bivariate substitution relates the flag Hilbert--Poincar\'e series of a
hyperplane arrangement $\A$ with the motivic zeta function $Z_{M(\A)}(Y,T)$
introduced in~\cite[Definition~1.1]{JKU/21} associated with the (representable)
matroid $M(\A)$ determined by~$\A$.


Second, we discuss in Section~\ref{subsec:Igusa.intro} an alternative
combinatorial formula (see Theorem~\ref{thm:atom.braid}) for specific
multivariate substitutions, viz.\ \emph{atom zeta functions}, associated with
\emph{classical} Coxeter arrangements --- viz.\ irreducible Coxeter arrangements
of types $\mathsf{A}$, $\mathsf{B}$, or~$\mathsf{D}$ --- in terms of total
partitions and rooted trees.

Third, we focus in Section~\ref{subsec:coarse.intro} on \emph{coarse flag
  Hilbert--Poincar\'e series}, viz.\ the bivariate ``coarsening'' of the flag
Hilbert--Poincar\'e series $\fHP_{\A}(Y,\bfT)$ obtained by setting $T_x=T$ for
all~$x$. Our Theorem~\ref{thm:coarse.Y=1} presents coarse flag
Hilbert--Poincar\'e series associated with Coxeter arrangements as
``$Y$-analogs'' of Hilbert series of the Stanley--Reisner rings of the first
barycentric subdivisions of standard simplices. On the level of rational
generating functions, this is reflected by an intriguing connection with
Eulerian polynomials.

\medskip

We record some examples of the rational functions $\fHP_{\A}$ in
Section~\ref{sec:exa}. The case of Boolean arrangements $\A = \msfA_1^n$ (see
Section~\ref{subsec:Boolean}) is of particular interest: specific
substitutions of the functions $\fHP_{\msfA_1^n}$ arise in the study
\cite{RV:CICO} of the average sizes of kernels of generic matrices with
support constraints over finite quotients of compact discrete valuation rings;
see \cite[Section~4.8]{MV/21}.



\section{Flag Hilbert--\texorpdfstring{Poincar\'e}{Poincare} series and \texorpdfstring{$\mathfrak{p}$}{p}-adic integrals} 
\label{subsec:Igusa.intro}

For general arrangements $\A$ over fields of characteristic zero, the functions 
$\fHP_{\A}$ are universal objects from
which various $\mfp$-adic integrals associated with $\A$ may be
obtained via specializations. To discuss this connection, we first
recall some representability properties of hyperplane arrangements.

If $\nofield$ is a field and $\A_{\nofield}$ is a hyperplane arrangement
defined over $\nofield$ such that $\intpos(\A)\cong\intpos(\A_{\nofield})$ as
posets, then we say that $\A$ is \emph{$\nofield$-representable} and call
$\A_{\nofield}$ a \emph{$\nofield$-representation} of~$\A$.  If, as we now
assume, $\A$ is a hyperplane arrangement defined over a field $\field$ of
characteristic zero, there exists a finite extension $\nofield$ of
$\mathbb{Q}$ such that $\A$ is $\nofield$-representable;
\textit{cf.}~\cite[Proposition~6.8.11]{Oxley}. Having fixed such a $\nofield$-representation
of~$\A$, we may further assume, without loss of generality, that each
$H\in \A$ is of the form $H=V(L)$, where
$L(\bfX) = c_L+\sum_{j=1}^d\alpha_{L,j}X_j\in\Gri_{\nofield}[\bfX]$ is an
affine linear polynomial over $\Gri_{\nofield}$, the ring of integers of the
number field~$\nofield$.  These choices allow us, in fact, to identify the
arrangement $\A$ with the collection of polynomials $L$ arising in this
way. We will use this freedom frequently.

In the sequel we denote by $\lri$ a compact discrete valuation ring (cDVR)
with an $\Gri_{\nofield}$-module structure. This could be a finite extension
of the completion $\Gri_{\nofield,\mfp}$ of $\Gri_{\nofield}$ at a nonzero
prime ideal $\mfp$ (in characteristic zero) or a power series ring of the form
$\Fq\llbracket X \rrbracket$, where $\Fq$ is the residue field of such a ring
(in positive characteristic).

Denoting by $\mfp$ the unique maximal ideal of $\lri$, we write
$\A(\lri/\mfp)$ for the reduction of $\A$ modulo~$\mfp$. If
$\mcL(\A)\cong \mcL(\A(\lri/\mfp))$, then $\A$ is said to have \emph{good
  reduction over~$\Fq$}, provided $\lri/\mfp$ has cardinality~$q$. It is
well-known that $\A$ has good reduction over $\Fq$ for all such $q$ not
divisible by finitely many (``bad'') primes; \textit{cf.}~\cite[Chap.~5.1]{Stanley/07}.

We now explain the connection between the flag Hilbert--Poincar\'e
series $\fHP_{\A}$ and various (multi- and univariate) $\mfp$-adic
integrals associated with the hyperplane arrangement~$\A$.

\begin{defn}\label{def:analytic.zf} 
    The \emph{analytic zeta function} of $\A$ over $\lri$ is
    \begin{align*}
        \zeta_{\A(\lri)}(\bfs) &= \int_{\lri^{\dim(\A)}}
        \prod_{x\in\toplat(\A)} \norm{\A_x}^{s_x} \, |\textup{d}\bfX|,
    \end{align*} 
    where $s_x$ is a complex variable for each $x\in\toplat(\A)$,
    further $\norm{\mathcal{X}} := \max\{|f| ~|~ f\in \mathcal{X}\}$
    for a finite set $\mathcal{X}\subset\lri$, where $|f|$ is the
    $\mfp$-adic absolute value on $\lri$, and $|\textup{d}\bfX|$ is
    the additive Haar measure on $\lri^{\dim(\A)}$, normalized so
    that~$\lri^{\dim(\A)}$ has measure~$1$.
\end{defn}

\begin{ex}\label{ex:zeta-A2}
  The analytic zeta function of $\mathsf{A}_2$, \textit{cf.}\ Example~\ref{ex:fHP-A2}, 
  over $\lri$ is 
  \begin{align*} 
    \zeta_{\mathsf{A}_2(\lri)}(\bfs) &= \int_{\lri^3} |X_1-X_2|^{s_{12}}|X_1-X_3|^{s_{13}}|X_2-X_3|^{s_{23}}\norm{X_1-X_2, X_1-X_3, X_2-X_3}^{s_{\widehat{1}}} |\textup{d}\bfX| \\
    &= \left. \dfrac{1}{1 - q^{-2-s_{12}-s_{13}-s_{23}-s_{\widehat{1}}}} \middle((1-q^{-1})(1-2q^{-1}) \right. \\
    &\hspace{113pt} \left. + (1-q^{-1})^2 \left(\dfrac{q^{-1-s_{12}}}{1-q^{-1-s_{12}}} + \dfrac{q^{-1-s_{13}}}{1-q^{-1-s_{13}}} + \dfrac{q^{-1-s_{23}}}{1-q^{-1-s_{23}}}\right)\right) .
  \end{align*} 
\end{ex} 

One sees the striking similarity between the rational functions in 
Example~\ref{ex:fHP-A2} and Example~\ref{ex:zeta-A2}.  Our next main result 
establishes that the functions $\zeta_{\A(\lri)}(\bfs)$ and $\fHP_{\A}(Y,\bfT)$ 
determine each other, in the following precise sense.

\begin{thmabc}\label{thm:equivalence} 
    Let $\A$ be a hyperplane arrangement over a number
    field~$\nofield$. For indeterminates
    $\bfs:=(s_x)_{x\in\toplat(\A)}$ and
    $\bm{r}:=(r_x)_{x\in\toplat(\A)}$ and $x\in\toplat(\A)$, let
    \begin{align*}%\label{def:gxs} 
        g_x(\bfs) &= \codim(x) + \sum_{y\in \toplat(\A_x)}s_y,
       \\ h_x(\bm{r}) &= \sum_{y\in \toplat(\A_x)} (r_y -
        \codim(y))\mu(y, x).%\label{def:hxr}
    \end{align*} 
    If $\lri$ is a cDVR and an $\Gri_{\nofield}$-module with residue
    field cardinality $q$ such that $\A$ has good reduction
    over~$\Fq$, then
    \begin{align*} 
        \zeta_{\A(\lri)}\left(\bfs\right) &=
        \fHP_{\A}\left(-q^{-1},\left(q^{-g_x(\bfs)}\right)_{x\in
                                            \toplat(\A)}\right), %\label{eqn:equiv1}
      \\ \fHP_{\A}\left(-q^{-1},
        (q^{-r_x})_{x\in\toplat(\A)}\right) &=
        \zeta_{\A(\lri)}\left(\left(h_x(\bm{r})\right)_{x\in
          \toplat(\A)}\right).%\label{eqn:equiv2}
    \end{align*}
\end{thmabc}

The interpretation of $\fHP_{\A}$ in terms of
the $\mfp$-adic integrals $\zeta_{\A(\lri)}$ expressed by
Theorem~\ref{thm:equivalence} is key to our proof of
Theorem~\ref{thm:reciprocity.new}.


\section{Igusa and atom zeta functions}
\label{subsec:Igusa-topological}

Assume now, as in Section~\ref{subsec:Igusa.intro}, that $\A$ is a
$\nofield$-representation and $\lri$ is a cDVR and an $\Gri_{\nofield}$-module. An important specialization of the multivariate function
$\zeta_{\A(\lri)}\left(\bfs\right)$ yields the (univariate)
\emph{Igusa local zeta function} (over~$\lri$) associated with the
product $f_{\A}(\bfX) := \prod_{L \in \A}L(\bfX)$ of linear polynomials
$L\in\Gri_{\nofield}[\bfX]$ (see \cite{Denef:Report}):
\begin{equation}\label{eqn:Igusa}
    \mathsf{Z}_{f_{\A},\lri}(s) := \zeta_{\A(\lri)}\left(
    \left(s\cdot\delta_{|\A_x|=1}\right)_{x\in\toplat(\A)}\right) =
    \int_{\lri^{\dim(\A)}}|f_{\A}|^s |\textup{d}\bfX|;
\end{equation}
here $s$ is a complex variable. Motivic zeta functions related with such
integrals have been studied and can be used to understand the topological zeta
function %$\mathsf{Z}_{f_{\A}}^{\mathrm{top}}(s)$
associated with~$f_{\A}(\bfX)$. This is executed, for example,
in~\cite{budur2020contact} for generic central arrangements, among others;
\textit{cf.}\ Section~\ref{subsec:gen.central.arr}. See Section~\ref{subsec:A3} for a
formula for Igusa's local zeta function associated with the braid
arrangement~$\mathsf{A}_3$.

% \section{Atom zeta functions}\label{subsec:atom.braid}

The specialization $\mathsf{Z}_{f_{\A},\lri}(s)$ defined
in~\eqref{eqn:Igusa} loses sight of all variables not corresponding to
atoms (\textit{i.e.}\ minimal elements in $\widetilde{\mcL}(\A)$) and cannot
distinguish atoms. We consider the following,
slightly more distinguishing $\mfp$-adic specialization
of~$\zeta_{\A(\lri)}(\bfs)$. 

\begin{defn}\label{def:atom} 
    The \emph{atom zeta function} of $\A$ is
    \begin{equation*}
        \zeta_{\A(\lri)}^{\atom}\left((s_L)_{L\in\A}\right) =
        \zeta_{\A(\lri)}\left((s_x\cdot
        \delta_{|\A_x|=1})_{x\in\toplat(\A)}\right) =
        \int_{\lri^{\dim(\A)}} \prod_{L\in\A} |L|^{s_L}\,
        |\textup{d}\bfX|.
      \end{equation*}
\end{defn}
Here we identified atoms with elements $L\in\A$. We note that the
independent variables $(s_L)_{L\in\A}$ allow for the treatment of
multi-arrangements in the sense of~\cite{budur2020contact}.

We remark that the atom zeta function is the finest coarsening of the
multivariate zeta function $\zeta_{\A(\lri)}(\bfs)$ that is, in
general, multiplicative with respect to direct products of hyperplane
arrangements. Namely, if $\A$ and $\A'$ are arrangements of
hyperplanes in (disjoint vector spaces) $K^d$ and $K^{d'}$ and $\lri$
is as above, then, by Fubini's theorem,
$$\zeta^{\atom}_{(\A\times\A')(\lri)}((\bfs,\bfs')) =
\zeta^{\atom}_{\A(\lri)}(\bfs)\zeta^{\atom}_{\A'(\lri)}(\bfs').$$

Our next result paraphrases an explicit combinatorial formula for atom zeta
functions associated with classical Coxeter arrangements;
\textit{cf.}~\cite[Theorem~5.6]{MV/21}. There we define, in particular, for $n\in\N$, the
sets $\TPnew{\mathsf{X}}{n}$ of total partitions of type~$\mathsf{X}_n$; for
type $\mathsf{X}=\mathsf{A}$, these are also defined
in~\cite[Example~5.2.5]{Stanley:Vol2} and related to Schr\"oder's fourth
problem.

\begin{thmabc}\label{thm:atom.braid}
  Let $\mathsf{X} \in \{\mathsf{A},\mathsf{B},\mathsf{D}\}$ and
  $n\in\N$, with $n\geq 2$ if~$\mathsf{X}=\mathsf{D}$. Then there
  exist, for all $\tau \in \TPnew{\mathsf{X}}{n}$, explicitly
  determined polynomials $\pitwo{X}{\tau}(Y)\in\Z[Y]$ and products of
  geometric progressions $\treeCgp_{\mathsf{X},\tau}\left( Z,
  (T_L)_{L\in\mathsf{X}_n}\right)$ such that the following holds: for
  all cDVR $\lri$ with residue field cardinality~$q$, assumed to be
  odd unless $\mathsf{X} = \mathsf{A}$,
    \begin{align*}
        \zeta_{\mathsf{X}_n(\lri)}^{\atom}\left((s_L)_{L\in\mathsf{X}_n}\right)
        &= \frac{1}{1-q^{-n-\sum_{L\in\mathsf{X}_n}s_L}}
        \sum_{\tau\in\TPnew{\mathsf{X}}{n}}\pitwo{X}{\tau}(-q^{-1})
        \treeCgp_{\mathsf{X},\tau}\left(q^{-1},(q^{-s_L})_{L\in
          \mathsf{X}_n}\right).
    \end{align*}
\end{thmabc}
  
% One may use these formulae to obtain alternative combinatorial formulae for
% the (multi- and univariate) topological zeta functions given in
% Corollary~\ref{cor:topological}. We leave the details to the reader and refer to Example~\ref{ex:A3}.

As a consequence of Theorem~\ref{thm:atom.braid}, we obtain an explicit formula
for Igusa's local zeta function $\mathsf{Z}_{f_{\mathsf{A}_n},\lri}(s)$ in terms
of (unlabeled) rooted trees with $n+1$ leaves; see Section~\ref{subsec:A3}.
%  In
% Corollary~\ref{cor:B.typeA} we express the atom zeta
% functions~$\zeta^{\textup{at}}_{\mathsf{B}_n(\lri)}(\bfs)$
% and~$\zeta^{\textup{at}}_{\mathsf{D}_n(\lri)}(\bfs)$ as sums over
% $\TPnew{A}{n}$, a set notably smaller than~$\TPnew{B}{n}$
% and~$\TPnew{D}{n}$.

\section{Coarse flag Hilbert--\texorpdfstring{Poincar\'e}{Poincare} series}\label{subsec:coarse.intro}

Consider now the bivariate specialization of the flag
Hilbert--Poincar\'e series~$\fHP_{\A}$ obtained by setting $T_x = T$
for each $x\in \toplat(\A)$:
\begin{defn}\label{def:coarse.new}
    The \emph{coarse flag Hilbert--Poincar\'e series} of $\A$ is
    $$\cfHP_{\A}(Y,T) = 
    \sum_{F\in\Delta(\toplat(\A))} \pi_F(Y)
    \left(\dfrac{T}{1 - T}\right)^{|F|}\in\Q(T)[Y].$$
\end{defn}

We define the polynomial $\mathcal{N}_{\A}(Y, T)\in\Q[Y,T]$ by the
formula
\begin{equation*}%\label{def:numer.coarse}
    \cfHP_{\A}(Y,T) =
    \frac{\mathcal{N}_{\A}(Y,T)}{(1-T)^{\rank(\A)}}.
\end{equation*}
We explore a number of remarkable properties
of these rational functions, including nonnegativity features of
$\mcN_{\A}(Y,T)$ and --- in the case of Coxeter
arrangements --- connections with Eulerian and Stirling numbers.

In Proposition~\ref{prop:coarse.Y=0}, we observe that $\mcN_{\A}(0,T)$
has nonnegative coefficients. Its proof is based on the fact that
$\cfHP_{\A}(0,T)$ is the coarse Hilbert series of the Stanley--Reisner
ring of the order complex of $\toplat(\A)$. The Cohen--Macaulayness of
this complex implies the nonnegativity of the associated $h$-vector,
\textit{i.e.}~the coefficients of~$\mathcal{N}_{\A}(0, T)$.

Recall that the $n$th \emph{Eulerian polynomial} $E_n(T)$ is defined via
\begin{equation*}%\label{def:euler.poly}
  E_n(T) = \sum_{w\in S_n}T^{\mathrm{des}(w)} \in \Z[T],
\end{equation*}
where $\des(w) := |\{i\in [n-1] ~|~ w(i) > w(i+1)\}|$.  Let $S(n,k)$
be the Stirling number of the second kind; see
\cite[Section~1.9]{Stanley:Vol1}.  It is well-known that
\begin{equation}\label{def:hilbert.series}
  \frac{E_n(T)}{(1-T)^n} = \sum_{k=1}^{n}k!\;S(n,k)
  \left(\frac{T}{1-T}\right)^{k-1}
\end{equation}
is the (coarse) Hilbert series of the Stanley--Reisner ring
$\F[\sd(\partial \Delta_{n-1})]$ associated with the first barycentric
subdivision of the boundary of the $(n-1)$-dimensional simplex
$\Delta_{n-1}$ over a field~$\F$;
\textit{cf.}~\cite[Theorem~9.1]{Petersen/15}. %Recall that the
%\emph{Stanley--Reisner ring} $\F[\Delta]$ of a simplicial complex
%$\Delta$ on $[m]$ is the quotient of the polynomial ring
%$\F[X_1,\dots, X_m]$ by the ideal $(\bm{X}_\sigma ~|~
%\sigma\notin\Delta)$, where $\bm{X}_{\sigma}=\prod_{i\in \sigma} X_i$;
%see~\cite[Chap.~II]{Stanley:commutative}.

A real hyperplane arrangement $\A$ is a \emph{Coxeter arrangement} if
the set of reflections across its hyperplanes fixes $\A$ and forms a
finite Coxeter group under composition. We call $\A$
\emph{irreducible} if it is not a direct product of two nontrivial
arrangements. Finite Coxeter arrangements may be decomposed as direct
products of {irreducible} Coxeter arrangements. The latter come in two
classes: \emph{classical} Coxeter arrangements of types $\mathsf{A}$,
$\mathsf{B}$, or $\mathsf{D}$ and \emph{exceptional} Coxeter
arrangements of types $\mathsf{E}_6$, $\mathsf{E}_7$, $\mathsf{E}_8$,
$\mathsf{F}_4$, $\mathsf{G}_2$, $\mathsf{H}_2$, $\mathsf{H}_3$,
$\mathsf{H}_4$, or $\mathsf{I}_{2}(m)$ for $m\geq 7$.

The following result shows that the coarse flag Hilbert--Poincar\'e series of
(most) Coxeter arrangements may be viewed as ``$Y$-analogs'' of the Hilbert
series~\eqref{def:hilbert.series}.

\begin{thmabc}\label{thm:coarse.Y=1}
    Let $\A$ be a Coxeter arrangement with no irreducible factor
    equivalent to~$\mathsf{E}_8$ and $\F$ be a field. Then
    \begin{equation}\label{eqn:Eulerian.coarse}
        \dfrac{\cfHP_{\A}(1, T)}{\pi_{\A}(1)} =
        \frac{E_{\rk(\A)}(T)}{(1-T)^{\rk(\A)}} = \Hilb(\F[\sd(\partial
          \Delta_{\rk(\A)-1})], T).\end{equation}
    In other words,
\[
\mathcal{N}_{\A}(1,T) = \pi_{\A}(1) E_{\rk(\A)}(T)
\]
and, equivalently, for $1 \leq k \leq \rk(\A)$,
        \begin{align*} 
      \sum_{\substack{F\in\Delta(\proplat(\A)) \\ |F| = k-1}}
      \dfrac{\pi_F(1)}{\pi_{\A}(1)} &= k! \; S(\rk(\A),k).
    \end{align*} 
\end{thmabc}

The Stirling numbers of the second kind enter our proof of
Theorem~\ref{thm:coarse.Y=1} via a simple formula, essentially due to Cayley,
for the numbers of plane trees of a given length and number of leaves.

Simple examples show that the conclusion of Theorem~\ref{thm:coarse.Y=1} does
not hold for general, non-Coxeter hyperplane arrangements, even when they are
central: the coefficients of $\mathcal{N}_{\A}(1,T)$ are typically not multiples
of~$\pi_{\A}(1)$. However, equation~\eqref{eqn:Eulerian.coarse} of
Theorem~\ref{thm:coarse.Y=1} holds for small-rank non-Coxeter restrictions of
type-$\mathsf{D}$ arrangements. For a recent strengthening of
Theorem~\ref{thm:coarse.Y=1} see~\cite{KM/21}, where they show that a real
central hyperplane arrangement satisfies~\eqref{eqn:Eulerian.coarse} if and only
if all of its chambers (\textit{i.e.}~the open regions in $\mathbb{R}^d$ not contained in
any hyperplane) are simplicial cones.

% \begin{quest}
%     Which further hyperplane arrangements
%     satisfy~\eqref{eqn:Eulerian.coarse}? Which property of hyperplane
%     arrangements does this equation reflect?
%     \end{quest}

To prove Theorem~\ref{thm:coarse.Y=1}, we first reduce to the irreducible case
by showing that coarse flag Hilbert--Poincar\'e series are, essentially,
Hadamard multiplicative; see \cite[Proposition~6.3]{MV/21}. For
$\mathsf{X}\in\{\mathsf{A}, \mathsf{B}, \mathsf{D}\}$, the result is proven,
type-by-type, by constructing a bijection between flags of
$\intpos(\mathsf{X}_n)$ and certain labeled rooted trees. We computed the
coarse flag Hilbert--Poincar\'e series of the other irreducible Coxeter
arrangement with the help of \textsc{HypIgu}~\cite{hypigu}, a
\SageMath~\cite{sagemath} package developed by the first author to compute
(coarse) flag Hilbert--Poincar\'e series and other rational functions
associated with hyperplane arrangements. The results of these computations,
along with many other examples, are recorded in the appendix of~\cite{MV/21};
in each case, the validity of Theorem~\ref{thm:coarse.Y=1} follows by
inspection. The type $\mathsf{E}_8$ is excluded from
Theorem~\ref{thm:coarse.Y=1} only because we do not supply a proof nor an
explicit computation.

All our computations support the following general nonnegativity conjecture.
\begin{conabc}\label{conj}
  For all hyperplane arrangements $\A$, the
  polynomial $\mathcal{N}_{\A}(Y,T)\in\Z[Y,T]$ has nonnegative
  coefficients.
\end{conabc}

Indeed, the polynomial $\mathcal{N}_{\A}(Y, T)$ has nonnegative coefficients
for all of the arrangements in the appendix of~\cite{MV/21}; see
Section~\ref{subsec:nonstrike} for the arrangement~$\mathsf{E}_6$. It is shown
in~\cite{KM/21} that Conjecture~\ref{conj} holds for all rank $3$ central
hyperplane arrangements. We furthermore view Conjecture~\ref{conj} as an
extension of the following observation, which uses deep results from algebraic
combinatorics. We note that $\pi_{\A}(Y)$ is the Poincar\'e polynomial of a
quotient of an exterior algebra, known as the Orlik--Solomon
algebra~\cite[Theorem~3.68]{OrlikTerao/92}.
\begin{prop} \label{prop:coarse.Y=0} For all hyperplane arrangements $\A$ we
  have $$\mathcal{N}_{\A}(Y, 0) = \pi_{\A}(Y)$$ and, for all fields
  $\mathbb{F}$,
  \begin{align*} 
    \dfrac{\cfHP_{\A}(0, T)}{1 - T} &=
    \Hilb(\mathbb{F}[\Delta(\intpos(\A))], T).
  \end{align*}
  In particular, the coefficients of both $\mathcal{N}_{\A}(Y, 0)$ and $\mathcal{N}_{\A}(0, T)$ are nonnegative.
\end{prop}


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

We discuss various examples of flag Hilbert--Poincar\'e series.


% \section{Arrangements of $m$ lines through the origin: $\mathsf{I}_2(m)$}\label{subsec:I2m}

% Let $m\geq 2$ and $\zeta_m$ be a primitive $m$th root of unity. We consider the
% Coxeter arrangement $\A=\{H_1,\dots,H_m\}$ of type~$\msfI_2(m)$, comprising $m$
% distinct lines $H_i$ through the origin in~$K^2$, where $K=\Q(\zeta_m)$. Let
% $\A$ be set of the $m$ linear factors of the polynomial $X_1^m+X_2^m\in
% K[\bfX]$. The flag Poincar\'e polynomials are
% \begin{align*}
%     \pi_\varnothing(Y) &= (1+Y)(1+(m-1)Y), & \pi_{(H_i)}(Y) &=
%     (1+Y)^2.
% \end{align*}

% With these data, it is easy to deduce the following.
% \begin{prop}\label{pro:Im} For $m\geq 2$,
%     \begin{align*} 
%       \fHP_{\msfI_2(m)}(Y,\bfT) &= \frac{1 + Y}{1-T_{\widehat{1}}} \left(1+(m-1)Y + (1+Y)\sum_{i=1}^m \gp{T_i}\right).
%     \end{align*}
% \end{prop}


\subsection{Boolean arrangements}
\label{subsec:Boolean} 

Consider the arrangement $\A$ comprising all the coordinate
hyperplanes in $\field^n$, also known as the Boolean arrangement and
equivalent to~$\mathsf{A}_1^n$.  Since $\A_x^y$ is Boolean for all
$x,y\in\intpos(\A)$ with $y<x$, it follows that $\pi_F(Y)=(1+Y)^n$ for
all $F\in \Delta(\toplat(\A))$. Thus,
\begin{equation*}\label{eqn:alg.boolean}
    \fHP_{\mathsf{A}_1^n}(Y,\bfT) = (1+Y)^n \sum_{F\in
      \Delta(\toplat(\A))} \prod_{x\in F}\gp{T_x}.
\end{equation*}
We identify $\Delta(\toplat(\A))$ with $\widetilde{\textup{WO}}([n])$,
the poset of chains of nonempty subsets of $[n]$ and rewrite
$\fHP_{\A}(Y,(T_I)_{I\in \mcP(n)\setminus\{\varnothing\}})$ in terms of
the \emph{weak order zeta function} (cf.~\cite[Definition~2.9]{SV1/15}) 
\begin{equation*}%\label{eqn:WO}
  \mathcal{I}^{\textup{WO}}_n\left((T_I)_{I\in
    \mcP(n)\setminus\{\varnothing\}}\right) := \sum_{F\in
    \widetilde{\textup{WO}}([n])}\prod_{I\in F} \gp{T_I};
\end{equation*} 

\begin{prop}\label{prop:boolean} 
    For $n\geq 1$,
    \begin{align*}
      \fHP_{\mathsf{A}_1^n}(Y,\bfT) &= (1+Y)^n \mathcal{I}^{\textup{WO}}_n(\bfT).
    \end{align*}
\end{prop}


The neat factorization of $\fHP_{\mathsf{A}_1^n}(Y,\bfT)$ as a product
of a polynomial in $Y$ and a rational function in $\bfT$ seems to be
atypical for these series. We have not observed such a factorization
anywhere outside the family of Boolean arrangements.


\subsection{Generic central arrangements}\label{subsec:gen.central.arr}

Let $m,n\in\N$ with $n\leq m$. We consider the arrangement $\mathcal{U}_{n,m}$
of $m$ hyperplanes through the origin in $\field^n$ in general position. That
is, for each $k\leq n$, every $k$-set of hyperplanes intersects in a
codimension-$k$ subspace. This is also known as the $m$-element uniform matroid
of rank~$n$. Observe that $\mathcal{U}_{n, n}$ is Boolean, seen in
Section~\ref{subsec:Boolean}.

For a set $I$, denote by $\mathcal{P}(I)$ the power set of
$I$ and by $\mathcal{P}(I; k)$ the set of subsets of $I$ of
cardinality~$k$. 
%We augment our power-set notation to accommodate multiple, prescribed
%subset sizes.
For $I,J\subset \N_0$, we define
$\mcP(I;J) = \bigcup_{j\in J} \mcP(I; j) \subseteq \mcP(I)$. Observe that for
$n\leq m$ we have $\proplat(\mathcal{U}_{n,m})\cong \mathcal{P}([m]; [n-1])$
and $\toplat(\mathcal{U}_{n,n}) \cong \mathcal{P}([n]; [n])$.

The next proposition generalizes
Proposition~\ref{prop:boolean}, as~$\mathsf{A}_1^n \cong
\mathcal{U}_{n,n}$.

\begin{prop}\label{prop:uniform-zeta}
Let $m,n\in\N$ with $n\leq m$. For $\bfT=(T_{\widehat{1}}, (T_I)_{I\in
\mcP([m]; [n-1])})$, 
    \begin{align*} 
        \fHP_{\mathcal{U}_{n,m}}\left(Y, \bfT\right) &= \dfrac{1 + Y}{1 - T_{\widehat{1}}}\sum_{k=0}^{n-1}\binom{m-1}{k}Y^k +
        \sum_{\ell=1}^{n-1}\sum_{k=0}^{n-\ell-1} \dfrac{(1+Y)^{\ell+1}}{1 - T_{\widehat{1}}} \binom{m-\ell-1}{k}Y^k \\
        &\quad \times \sum_{I\in \mcP([m]; \ell)}  T_I
        \mathcal{I}^{\textup{WO}}_\ell\left((T_J)_{J\in \mcP(I;
        [\ell])}\right).
    \end{align*} 
\end{prop}


\subsection{Braid arrangement \texorpdfstring{$\mathsf{A}_3$}{A3}}\label{subsec:A3}

Our work allows us to obtain explicit formulae for the Igusa zeta functions
associated with the braid arrangements of type $\mathsf{A}$ in terms of rooted
trees. Consider, for example, the braid arrangement $\mathsf{A}_3$, with
$f_{\mathsf{A}_3}(\bfX) = \prod_{1\leq i < j \leq 4}(X_i - X_j)$. 
\begin{figure}[h]
    \centering
    \begin{subfigure}[b]{0.18\textwidth} 
        \centering
        \begin{tikzpicture}
            \pgfmathsetmacro{\x}{0.75} \pgfmathsetmacro{\y}{0.5}
            \node(1) at (0, \y) [circle, fill=black, inner sep=1.5pt]
                 {}; \node(2) at (-\x, 0) [circle, fill=black, inner
                   sep=1.5pt] {}; \node(3) at (-0.33*\x, 0) [circle,
                   fill=black, inner sep=1.5pt] {}; \node(4) at
                 (0.33*\x, 0) [circle, fill=black, inner sep=1.5pt]
                 {}; \node(5) at (\x, 0) [circle, fill=black, inner
                   sep=1.5pt] {}; 
                 \draw[-] (1) -- (2); \draw[-] (1) -- (3); \draw[-]
                 (1) -- (4); \draw[-] (1) -- (5);
        \end{tikzpicture}
    \end{subfigure}~%
    \begin{subfigure}[b]{0.18\textwidth} 
        \centering
        \begin{tikzpicture}
            \pgfmathsetmacro{\x}{0.75} \pgfmathsetmacro{\y}{0.5}
            \node(1) at (0, 2*\y) [circle, fill=black, inner
              sep=1.5pt] {}; \node(2) at (-0.66*\x, \y) [circle,
              fill=black, inner sep=1.5pt] {}; \node(3) at (0, \y)
                 [circle, fill=black, inner sep=1.5pt] {}; \node(4) at
                 (0.66*\x, \y) [circle, fill=black, inner sep=1.5pt]
                 {}; \node(5) at (-1*\x, 0) [circle, fill=black, inner
                   sep=1.5pt] {}; \node(6) at (-0.33*\x, 0) [circle,
                   fill=black, inner sep=1.5pt] {}; \draw[-] (1) -- (2); \draw[-] (1)
                 -- (3); \draw[-] (1) -- (4); \draw[-] (2) -- (5);
                 \draw[-] (2) -- (6);
        \end{tikzpicture}
    \end{subfigure}~%
    \begin{subfigure}[b]{0.18\textwidth} 
        \centering
        \begin{tikzpicture}
            \pgfmathsetmacro{\x}{0.75} \pgfmathsetmacro{\y}{0.5}
            \node(1) at (0, 2*\y) [circle, fill=black, inner
              sep=1.5pt] {}; \node(2) at (-0.66*\x, \y) [circle,
              fill=black, inner sep=1.5pt] {}; \node(3) at (0.66*\x,
            \y) [circle, fill=black, inner sep=1.5pt] {}; \node(4) at
            (-1*\x, 0) [circle, fill=black, inner sep=1.5pt] {};
            \node(5) at (-0.33*\x, 0) [circle, fill=black, inner
              sep=1.5pt] {}; \node(6) at (\x, 0) [circle, fill=black,
              inner sep=1.5pt] {}; \node(7) at (0.33*\x, 0) [circle,
              fill=black, inner sep=1.5pt] {}; \draw[-] (1) -- (2); \draw[-] (1) -- (3);
                 \draw[-] (1) -- (3); \draw[-] (2) -- (4); \draw[-]
                 (2) -- (5); \draw[-] (3) -- (6); \draw[-] (3) -- (7);
        \end{tikzpicture}
    \end{subfigure}~%
    \begin{subfigure}[b]{0.18\textwidth} 
        \centering
        \begin{tikzpicture}
            \pgfmathsetmacro{\x}{0.75} \pgfmathsetmacro{\y}{0.5}
            \node(1) at (0, 2*\y) [circle, fill=black, inner
              sep=1.5pt] {}; \node(2) at (-0.5*\x, \y) [circle,
              fill=black, inner sep=1.5pt] {}; \node(3) at (0.5*\x,
            \y) [circle, fill=black, inner sep=1.5pt] {}; \node(4) at
            (-1*\x, 0) [circle, fill=black, inner sep=1.5pt] {};
            \node(5) at (-0.5*\x, 0) [circle, fill=black, inner
              sep=1.5pt] {}; \node(6) at (0, 0) [circle, fill=black,
              inner sep=1.5pt] {};
            \draw[-] (1) -- (2); \draw[-] (1) -- (3); \draw[-] (2) --
            (4); \draw[-] (2) -- (5); \draw[-] (2) -- (6);
        \end{tikzpicture}
    \end{subfigure}~%
    \begin{subfigure}[b]{0.18\textwidth} 
        \centering
        \begin{tikzpicture}
            \pgfmathsetmacro{\x}{0.75} \pgfmathsetmacro{\y}{0.5}
            \node(1) at (0, 3*\y) [circle, fill=black, inner
              sep=1.5pt] {}; \node(2) at (-0.5*\x, 2*\y) [circle,
              fill=black, inner sep=1.5pt] {}; \node(3) at (0.5*\x,
            2*\y) [circle, fill=black, inner sep=1.5pt] {}; \node(4)
            at (-1*\x, \y) [circle, fill=black, inner sep=1.5pt] {};
            \node(5) at (0, \y) [circle, fill=black, inner sep=1.5pt]
                 {}; \node(6) at (-1.5*\x, 0) [circle, fill=black,
                   inner sep=1.5pt] {}; \node(7) at (-0.5*\x, 0)
                 [circle, fill=black, inner sep=1.5pt] {}; \draw[-] (1) -- (2); \draw[-]
                 (1) -- (3); \draw[-] (2) -- (4); \draw[-] (2) -- (5);
                 \draw[-] (4) -- (6); \draw[-] (4) -- (7);
        \end{tikzpicture}
    \end{subfigure}
    \caption{Five rooted trees with four leaves.}
    \label{fig:rooted-trees-4}
\end{figure}

By \cite[Corollary~5.7]{MV/21}, for all cDVR $\lri$ with residue field cardinality~$q$,
Igusa's zeta function associated with $f:=f_{\mathsf{A}_3}$ is
\begin{align*} 
    \mathsf{Z}_{f,\lri}(s) &= \left.\dfrac{1 -
      q^{-1}}{1-q^{-3 - 6s}}\middle((1-2q^{-1})(1-3q^{-1}) \right. \\  
    &\quad +
    \dfrac{6q^{-1-s}(1-q^{-1})(1-2q^{-1})}{1 - q^{-1-s}} + \dfrac{3q^{-2-2s}(1-q^{-1})^2}{(1 - q^{-1-s})^2} \\
    &\quad \left. +
    \dfrac{4q^{-2-3s}(1-q^{-1})(1-2q^{-1})}{1 - q^{-2-3s}} + 
    \dfrac{12q^{-3-4s}(1-q^{-1})^2}{(1-q^{-1-s})(1 - q^{-2-3s})} \right).
\end{align*}
The five summands correspond to the five rooted trees in
Figure~\ref{fig:rooted-trees-4}. Each summand is determined from combinatorial
data readily read off from the corresponding tree.

\subsection{Coarse flag Hilbert--\texorpdfstring{Poincar\'e}{Poincare} series}\label{subsec:nonstrike}

In the appendix of \cite{MV/21} we record the coarse flag Hilbert--Poincar\'e
series of various hyperplane arrangements, including the irreducible Coxeter
arrangements of rank oat most seven. Using our package \textsc{HypIgu} we
computed, for instance, the numerator of the coarse flag Hilbert--Poincar\'e
series of the arrangement $\mathsf{E}_6$:

\begin{align*}
    \mathcal{N}_{\mathsf{E}_6}(Y, T) &= 1 + 36Y + 510Y^2 + 3600Y^3 + 13089Y^4 + 22284Y^5 + 12320Y^6 \\
    &\quad + (4591 + 57420Y + 289824Y^2 + 748080Y^3 + 1020819Y^4 \\
    &\quad + 671940Y^5 + 162206Y^6)T + (103681 + 888840Y + 3011919Y^2 \\
    &\quad + 5080320Y^3 + 4411839Y^4 + 1858680Y^5 + 300401Y^6)T^2 \\
    &\quad + (300401 + 1858680Y + 4411839Y^2 + 5080320Y^3 + 3011919Y^4 \\
    &\quad + 888840Y^5 + 103681Y^6)T^3 + (162206 + 671940Y + 1020819Y^2 \\
    &\quad + 748080Y^3 + 289824Y^4 + 57420Y^5 + 4591Y^6)T^4 + (12320 \\
                                     &\quad + 22284Y + 13089Y^2 + 3600Y^3 + 510Y^4 + 36Y^5 + Y^6)T^5.
                                       \end{align*}


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\end{document}
