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\usepackage[utf8]{inputenc}
\usepackage{array}
\usepackage{amssymb,amsthm,amsmath,mathrsfs,amsfonts}
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\usepackage{amsmath,amssymb,amsthm}
\usepackage{amssymb, amsmath,amsthm,xcolor,bm,hyperref,color,tikz,euscript}
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\usepackage{tikz-cd}
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\usepackage[all]{xy}

\tikzstyle{invisivertex} = [black, shape=rectangle, minimum size=0pt, inner sep=3pt]
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\usepackage{cleveref}
\usepackage[all]{xy}

\usepackage[utf8]{inputenc}
\usepackage[english]{babel}
\usepackage[utf8]{inputenc}
\usepackage[english]{babel}
\usepackage{bbm}

% Abbreviations:

\newcommand{\ie}{{\textit{i.e.}}}

\newcommand{\cA}{\mathcal{A}}
\newcommand{\cK}{\mathcal{K}}
\newcommand{\cL}{\mathcal{L}}
\newcommand{\cI}{\mathcal{I}}
\newcommand{\cV}{\mathcal{V}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\cC}{\mathcal{C}}
\newcommand{\cD}{\mathcal{D}}
\newcommand{\cM}{\mathcal{M}}
\newcommand{\idInt}{\mathsf{IL}^{\ast}}
\newcommand{\Int}{\mathsf{IL}}
\newcommand{\R}{\mathbb{R}}

\newcommand{\aaa}{\mathbf{a}}
\newcommand{\bbb}{\mathbf{b}}
\newcommand{\ccc}{\mathbf{c}}
\newcommand{\ddd}{\mathbf{d}}

\newcommand{\Poin}{\mathcal{\sf Poin}}
\newcommand{\POIN}[3][]{\Poin_{#1}(#2; #3)} % \POIN[\cC]{P}{y} 
\newcommand{\sep}{{\sf sep}}
\newcommand{\edges}{{\sf edges}}
\newcommand{\peak}{{\sf Peak}}
\newcommand{\Des}{{\sf Des}}
\newcommand{\LinExt}{{\sf LinExt}}
\newcommand{\tuples}{{\sf Tup}}
\newcommand{\supp}{{\sf supp}}
\newcommand{\StatMon}{{\sf u}}
\newcommand{\StatMonNew}{{\sf U}}
\newcommand{\mon}{{\sf m}}
\newcommand{\incdec}{{\sf IncDec}}
\newcommand{\PIE}{{\sf A}}
\newcommand{\topPIE}{{\sf B}}
\newcommand{\coeffs}{{\sf C}}
\newcommand{\rank}{{\sf rank}}
\newcommand{\irank}{{\sf irank}}
\newcommand{\Rank}{{\sf Rank}}
\newcommand{\IRank}{{\sf IRank}}
\newcommand{\wt}{{\sf wt}}
\newcommand{\wtn}{\wt^+}
\newcommand{\wtp}{\wt^-}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Num}{{\sf Num}}
\newcommand{\TT}[2]{{\sf T}(#1, #2)} % \TT{\cM}{E}
\newcommand{\bout}{\operatorname{b-out}}


\newcommand{\multichain}[1]{\{\!\!\{#1\}\!\!\}}

\newcommand{\extPsi}{{\text{\scriptsize\sf ex}\hspace*{-1pt}}\Psi}
\newcommand{\pullPsi}{\Psi_{\sf pull}}
\newcommand{\combextPsi}{{\text{\scriptsize\sf cx}\hspace*{-1pt}}\Psi}
\newcommand{\combpullPsi}{\Psi_{\sf cpull}}
\newcommand{\combPsi}{\Psi_{\sf c}}

\newcommand{\QSym}{\textsf{QSym}}

\newcommand{\Dfn}[1]{\emph{\bfseries #1}}
%
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\newtheorem{note}[lemma]{Note}
\newtheorem{openproblem}[lemma]{Open Problem}
\newtheorem{example}[lemma]{Example}

\numberwithin{lemma}{section}

%% define your title in the usual way
\title{The Poincaré-extended $\aaa\bbb$-index}

%% 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}Department of This, University of That, There \\ \addressmark{2}School of Something, University of Something Else, Somewhere}

\author{Galen Dorpalen-Barry\thanks{\href{mailto:galen.dorpalen-barry@rub.de}{galen.dorpalen-barry@rub.de}}\addressmark{1}, Joshua Maglione\addressmark{2}, \and Christian Stump\addressmark{3}}

\address{\addressmark{1} University of Oregon, United States \\
\addressmark{2} University of Galway, Ireland\\
\addressmark{3} Ruhr-Universit\"at Bochum, Germany }

%% put the date of submission here
\received{\today}

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%\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{Motivated by a conjecture concerning Igusa local zeta functions for intersection posets of hyperplane arrangements, we introduce and study the \emph{Poincaré-extended $\aaa\bbb$-index}, which generalizes both the $\aaa\bbb$-index and the
Poincaré polynomial. For posets admitting $R$-labelings, we give a
combinatorial description of the coefficients of the extended
$\aaa\bbb$-index, proving their nonnegativity.
In the case of intersection posets of hyperplane arrangements,
we prove the above conjecture of the second author and Voll.
% This connection allows us to translate our results into the language of quasisymmetric functions, and---in the special case of symmetric functions---make a conjecture about Schur positivity.
}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{poset, matroid, oriented matroid, ab-index, hyperplane arrangement, R-labeling, quasisymmetric function}

%% 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
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\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
Grunewald, Segal, and Smith introduced the subgroup zeta function of finitely-generated groups~\cite{Grunewald-Segal-Smith}, and
Du Sautoy and Grunewald gave a general method to compute such zeta functions using $p$-adic integration and resolution of singularities~\cite{duSautoy-Grunewald}.
This motivated Voll and the second author to examine the setting where the multivariate polynomials factor linearly. They found that the $p$-adic integrals are specializations of multivariate rational functions depending only on the combinatorics of the corresponding hyperplane arrangement~\cite{Maglione-Voll}.
After a natural specialization, its denominator greatly simplifies, and they conjecture that the numerator polynomial has nonnegative coefficients.

In this work, we prove their conjecture, which is related to the poles of these zeta functions; see \Cref{rem:implications-for-zetas}.
Specifically, we reinterpret these numerator polynomials by introducing and studying the \emph{(Poincaré-)extended $\aaa\bbb$-index}, a polynomial generalizing both the \emph{Poincaré polynomial} and \emph{$\aaa\bbb$-index} of the \emph{intersection poset} of the arrangement.
These polynomials have been studied extensively in combinatorics, although from different perspectives.
The coefficients of the Poincaré polynomial have interpretations in terms of the combinatorics and the topology of the
arrangement~\cite[Section~2.5]{BjornerEtAl}.
The $\aaa\bbb$-index, on the other hand, carries information about the order complex of the poset and is
particularly well-understood in the case of face posets of oriented
matroids---or, more generally, Eulerian posets. In those settings, the
$\aaa\bbb$-index encodes topological data via the \emph{flag
$f$-vector}~\cite{bayer-survey}.

We study the extended $\aaa\bbb$-index in the generality of graded posets admitting $R$-labelings.
This class of posets includes intersection posets of hyperplane arrangements and, more generally, geometric lattices and geometric semilattices.
We show that the extended $\aaa\bbb$-index has nonnegative coefficients by interpreting them in terms of a combinatorial statistic.
This generalizes statistics given for the $\aaa\bbb$-index by Billera, Ehrenborg, and Readdy~\cite{billera-ehrenborg-readdy} and for the pullback $\aaa\bbb$-index (defined below) by Bergeron, Mykytiuk, Sottile and van Willigenburg~\cite{bergeron-mykytiuk-sottile-vanWilligenburg2000}.
This interpretation proves the aforementioned conjecture~\cite{Maglione-Voll}, as well as a related conjecture from Kühne and the second author~\cite{Kuhne-Maglione}.

Motivated by the proofs of these conjectures, we describe a close relationship between the Poincaré polynomial and the $\aaa\bbb$-index by showing that the extended $\aaa\bbb$-index can be obtained from the $\aaa\bbb$-index by a suitable substitution.
This recovers, generalizes and unifies several results in the literature.
Concretely, special cases of this substitution were observed by Billera, Ehrenborg and Readdy
%for $y=1$
for lattices of flats of \emph{oriented matroids}~\cite{billera-ehrenborg-readdy},
% by Bergeron, Mykytiuk, Sottile and van Willigenburg for general \emph{edge-labeled  graded posets}~\cite{bergeron-mykytiuk-sottile-vanWilligenburg2000},
by Saliola and Thomas for lattices of flats of \emph{oriented interval greedoids}~\cite{saliola-thomas}, and by Ehrenborg for \emph{distributive lattices}~\cite{Ehrenborg-signed}.
%Each of these concerns a pair of posets, related by an order- and rank- preserving map, generalizing to the relationship between the face poset and lattice of flats of a hyperplane arrangement.
% In agreement with those special cases, our substitution allows us to study the \emph{pullback $\aaa\bbb$-index}, a polynomial in noncommuting variables $\ccc = \aaa + \bbb$ and $2\ddd = 2(\aaa\bbb + \bbb\aaa)$ with connections to the combinatorics of \emph{$P$-partitions} and \emph{quasisymmetric functions} from~\cite{stembridge-p-partitions} and~\cite[Section~7]{bergeron-mykytiuk-sottile-vanWilligenburg2000}.
% We conjecture that our substitution (now defined on quasisymmetric functions) restricts to a map on symmetric function, and that the image of a Schur function is Schur positive.

% The remainder of this paper is organized as follows.
% In \Cref{sec:main-stuff}, we introduce the main definitions (\Cref{sec:main-defs}) and state the main results (\Cref{sec:mainresults}) of the paper.
% We then discuss in \Cref{sec:connections} how these recover, generalize and unify results in the context of $P$-partitions and quasisymmetric functions.
% The proofs are then presented in \Cref{sec:proof-of-combinatorial-interp,,sec:refinement-of-ber}.


\section{The Poincaré-Extended $\aaa\bbb$-index}
\label{sec:main-stuff}

\subsection{Main definitions}
\label{sec:main-defs}

Unless otherwise specified,~$P$ is a finite \Dfn{graded poset} of rank~$n$, that is, $P$ is a finite poset with unique minimum element~$\hat 0$ of rank~$0$ and unique maximum element~$\hat 1$ of rank~$n$ such that $\rank(X)$ is equal to the length of any maximal chain from~$\hat 0$ to $X$.
%We follow~\cite[Chapter 3]{ec1} for definitions and notation surrounding posets.
The \Dfn{M\"obius function}~$\mu$ of $P$ is given by $\mu(X, X) = 1$ for all $X\in P$ and $\mu(X, Y) = -\sum_{X\leq Z < Y} \mu(X, Z)$ for all $X < Y$ in $P$.
The \Dfn{Poincaré polynomial} of~$P$ is
\[
  \POIN{P}{y} = \sum_{X\in P} |\mu(\hat{0}, X)| \cdot y^{\rank(X)} \ \in \Z[y].
\]
The \Dfn{chain Poincaré polynomial} of a chain $\cC = \big\{\cC_1 < \dots < \cC_k \big\}$ in~$P\setminus\{\hat{1}\}$ is 
\[
  \POIN[\cC]{P}{y} =\prod_{i = 1}^k \POIN{[\cC_i,\cC_{i+1}]}{y} \ \in \Z[y], 
\]
where we set~$\cC_{k+1} = \hat{1}$. By taking the singleton chain $\{\hat 0\}$,
we recover the usual Poincaré polynomial, $\POIN{P}{y} = \POIN[\{\hat 0\}]{P}{y}$. The ranks of a given chain~$\cC$ is given by 
\[
  \Rank(\cC) = \left\{ \rank(\cC_i) ~\middle|~ 1 \leq i \leq k \right\}. 
\]
We often consider polynomials in noncommuting variables~$\aaa$ and~$\bbb$ with coefficients being polynomials in $\Z[y]$.
For a subset $S \subseteq \{i,i+1,\dots,j\}$, we write $\mon_S = m_i\dots m_j$ for the monomial with $m_k = \bbb$ if $k \in S$ and $m_k = \aaa$ if $k \notin S$ and we similarly write $\wt_S = w_i\dots w_j$ for the polynomial with
\begin{equation}
\label{eqn:ext-weights}
  w_k = \begin{cases}
          \bbb        & \text{if } k\in S, \\
          \aaa - \bbb & \text{if } k\notin S\,.
        \end{cases}
\end{equation}
The supersets $\{i,i+1,\dots,j\}$ are understood from the context as the set of all indices that can possibly be contained in the set~$S$.
In case of ambiguity, we in addition identify the considered superset.
For a chain $\cC$ in~$P$, we also set $\mon_\cC = \mon_{\Rank(\cC)}$ and $\wt_\cC = \wt_{\Rank(\cC)}$.
The following is the main object of study of this paper.

\begin{definition}
\label{def:extab}
  The (\Dfn{Poincaré-})\Dfn{extended $\aaa\bbb$-index} of~$P$ is
  \[
    \extPsi(P;y,\aaa,\bbb)
      = \sum_{\cC\text{ chain in } P\setminus\{\hat{1}\}} \POIN[\cC]{P}{y} \cdot \wt_{\cC} \ \in \Z[y]\langle\aaa,\bbb\rangle\,,
  \]
  where~$\wt_\cC = w_0 \cdots w_{n-1}$ is given in \Cref{eqn:ext-weights}.
\end{definition}

Since~$P$ has a unique minimum, we always have $\POIN{P}{0} = 1$, implying
\[
  \extPsi(P;0,\aaa,\bbb)
    = \sum_{\cC\text{ chain in } P\setminus\{\hat{1}\}} \wt_\cC\,.
\]
This recovers the \Dfn{$\aaa\bbb$-index}
\(
  \Psi(P;\aaa,\bbb) = \extPsi(P;0,\aaa,\bbb).
\)

\begin{example}
\label{ex:face-poset2}
  We compute the extended $\aaa\bbb$-index of the poset $\cL$ drawn below on the left.
  % On the right, we collect the relevant data.
  \begin{center}
    \begin{tikzpicture}[scale=1.2, baseline={([yshift=-.5ex]current bounding box.center)}]
      \node[invisivertex] (1)  at ( 0,2){$\hat 1$};
      \node[invisivertex] (H1) at (-1,1){$\alpha_1$};
      \node[invisivertex] (H2) at ( 0,1){$\alpha_2$};
      \node[invisivertex] (H3) at ( 1,1){$\alpha_3$};
      \node[invisivertex] (0)  at ( 0,0){$\hat 0$};
      \draw (0) -- (H1) -- (1);
      \draw (0) -- (H2) -- (1);
      \draw (0) -- (H3) -- (1);
    \end{tikzpicture}
     \hspace{2cm}
    \begin{tabular}{|c|c|c|c|}
      \hline
      &&&\\[-10pt]
      $\cC$ & $\POIN[\cC]{\cL}{y}$ & $\Rank(\cC)$ & $\wt_\cC$ \\[5pt]
      \hline
      &&&\\[-10pt]
      $\{\}$ & $1$ & $\{\}$ & $(\aaa-\bbb)^2$ \\[5pt]
      $\{\hat 0\}$ & $1 + 3y + 2y^2$ & $\{0\}$ & $\bbb(\aaa-\bbb)$ \\[5pt]
      $\{\alpha_i\}$ & $1+y$ & $\{1\}$ & $(\aaa-\bbb)\bbb$ \\[5pt]
      $\{\hat 0 < \alpha_i\}$ & $(1+y)^2$ & $\{0,1\}$ & $\bbb^2$\\
      \hline
    \end{tabular}
  \end{center}
  The extended $\aaa\bbb$-index and its specialization to the $\aaa\bbb$-index are thus
  \begin{align*}
    \extPsi(\cL;y,\aaa,\bbb)
      &= (\aaa-\bbb)^2\! +\! (1 + 3y + 2y^2)\bbb(\aaa-\bbb)\! +\! 3\cdot (1+y)(\aaa-\bbb)\bbb\! +\! 3\cdot(1+y)^2\bbb^2 \\
      &= \aaa^2 + (3y+2y^2)\bbb\aaa + (2+3y)\aaa\bbb + y^2\bbb^2, \\
    \Psi(\cL;\aaa,\bbb) &= \aaa^2 + 2\aaa\bbb\,.
  \end{align*}
\end{example}

\begin{remark}
  \label{rem:top-element-toss-out}
  Taking chains~$\cC$ in $P \setminus \{\hat 1\}$, rather than in~$P$, is a
  harmless reduction in the definition of the extended $\aaa\bbb$-index since
  $\POIN[\cC]{P}{y} = \POIN[\cC\cup\{\hat 1\}]{P}{y}$. If we consider both~$\cC$
  and~$\cC \cup\{\hat 1\}$ separately as summands of $\extPsi(P;y,\aaa,\bbb)$,
  we would need to consider weights $\wtn_\cC = w_0 \cdots w_n$
  taking also the $n$-th position into account.
  We would have the two terms $\POIN[\cC]{P}{y}\cdot\wtn_{\cC}$ and
  $\POIN[\cC \cup \{\hat 1\}]{P}{y}\cdot\wtn_{\cC \cup \{\hat 1\}}$,
  differing only in the last entry of the weight, so their sum is $\POIN[\cC]{P}{y}\cdot\wt_{\cC}\cdot \aaa$.
  This holds for all chains, proving
  \begin{equation}
    \extPsi(P;y,\aaa,\bbb)\cdot\aaa = \sum_{\cC\text{ chain in } P} \POIN[\cC]{P}{y} \cdot \wtn_\cC.
    \label{eq:enab01}
  \end{equation}
\end{remark}

% \begin{remark}
% \label{rem:bottom-element-toss-out}
%   In the case of the $\aaa\bbb$-index, a similar argument to the one in the
%   \Cref{rem:top-element-toss-out} implies that we could further restrict to
%   chains in $P\setminus\{\hat 0, \hat 1\}$. Therefore, we have 
%   \begin{equation}
%     \Psi(P;\aaa,\bbb) = \aaa\cdot\left(\sum_{\cC\text{ chain in } P\setminus\{\hat 0,\hat 1\}} \wtp_\cC\right),
%     \label{eq:ab01}
%   \end{equation}
%   where $\wtp_\cC = w_1 \dots w_{n-1}$ as neither~$0$ nor~$n$ can appear
%   in $\Rank(\cC)$. The expression in the parentheses recovers the usual
%   definition of the $\aaa\bbb$-index as given, for example, in~\cite[Section
%   2]{bayer-survey}. Using additional information, the extended $\aaa\bbb$-index
%   admits a similar property to that in \Cref{eq:ab01}, which is given in
%   \Cref{cor:enab0}.
% \end{remark}

The fact that $\hat 1$ is included in every chain in the computation of the
chain Poincaré polynomial is inspired by the setting of hyperplane arrangements; see~\cite{AguiarMahajan,OrlikTerao} for more details.
A (central, real) \Dfn{hyperplane arrangement} $\mathcal{A}$ is a finite collection of
hyperplanes in~$\mathbb{R}^d$, all of which have a
common intersection.
The \Dfn{lattice of flats} $\mathcal{L}$ of $\mathcal{A}$
is the poset of subspaces of $\mathbb{R}^d$ obtained from
intersections of subsets of the hyperplanes, ordered by reverse inclusion.
The open, connected components of the complement $\mathbb{R}^d\setminus \mathcal{A}$ are called
(open) \Dfn{chambers}.
The set of (closed) \Dfn{faces} $\Sigma$ is the set of \emph{closures} of chambers of $\cA$, together with all possible intersections of closures of chambers (ignoring intersections which are empty).
This set comes equipped with a natural partial order by reverse inclusion, and for this reason we refer to $\Sigma$ as the \Dfn{face poset} of $\mathcal{A}$.
There is an order-preserving, rank-preserving
surjection $\supp : \Sigma \twoheadrightarrow \cL$ sending a face to its affine
span~\cite[Proposition~4.1.13]{BjornerEtAl}. This map extends to chains, and the
fiber sizes are given, for $\cC = \{ \cC_1 < \dots < \cC_k \}\subseteq \cL$, by
\begin{equation}
  \#\supp^{-1}(\cC) = \prod_{i=1}^{k} \POIN{[\cC_i,\cC_{i+1}]}{1} = \POIN[\cC]{P}{1}, 
  \label{eq:supppre}
\end{equation}
with $\cC_{k+1} = \hat 1$; see~\cite[Proposition~4.6.2]{BjornerEtAl}.
This is the key motivation for the next definition.
\begin{definition}
\label{def:pullbackab}
  The \Dfn{pullback $\aaa\bbb$-index} of~$P$ is
  \[
    \pullPsi(P;\aaa,\bbb) = \extPsi(P;1,\aaa,\bbb).
  \]
\end{definition}

Let~$\Sigma$ be the face poset and $\cL$ the lattice of flats of a real central
hyperplane arrangement. Since~$\Sigma$ may not have a unique minimum element, we
formally add a minimum element~$\hat{0}$ and let $\Sigma \cup \{\hat 0 \}$ be
the resulting poset. Now, \Cref{eq:supppre} relates the $\aaa\bbb$-index of the face poset and the pullback $\aaa\bbb$-index of the lattice of flats by
\begin{equation}
  \Psi(\Sigma \cup\{\hat 0\};\aaa,\bbb) =  \aaa\cdot\pullPsi(\cL;\aaa,\bbb)\,.
  \label{eq:pullbackab}
\end{equation}
Note that this relates the evaluation of $\extPsi(\Sigma\cup\{\hat 0\};y,\aaa,\bbb)$ at $y=0$
to the evaluation of $\extPsi(\cL;y,\aaa,\bbb)$ at $y=1$.
\Cref{eq:supppre} and thus also \Cref{eq:pullbackab} hold indeed in the more general context of oriented
matroids.

\begin{example}
\label{ex:pullbackab}
  The pullback $\aaa\bbb$-index of the poset from \Cref{ex:face-poset2} is
  \[
    \pullPsi(\cL;\aaa,\bbb) = \extPsi(\cL;1,\aaa,\bbb) = \aaa^2 + 5\bbb\aaa + 5\aaa\bbb + \bbb^2\,.
  \]
  Consider the arrangement of three lines in the plane through a common intersection as shown below on the left in a way that emphasizes its face structure.
  Its lattice of flats is the poset $\cL$ from \Cref{ex:face-poset2}.
  To the right, we draw its face poset~$\Sigma$ with~$\hat 0$ included.
  \begin{center}
     \begin{tikzpicture}[scale=1.2]
       \draw[very thick] (  0:-2) -- (  0:2);
       \draw[thick] (120:-2) -- (120:2);
       \draw[thick] (240:-2) -- (240:2);
       \draw[white, fill=white]  (0,0) circle [radius=4pt];
       \draw[fill]  (0,0) circle [radius=2pt];
       \node[invisivertex] at (30:1){\tiny$+++$};
       \node[invisivertex] at (90:1){\tiny$-++$};
       \node[invisivertex] at (150:1){\tiny$--+$};
       \node[invisivertex] at (210:1){\tiny$---$};
       \node[invisivertex] at (270:1){\tiny$+--$};
       \node[invisivertex] at (330:1){\tiny$++-$};
      \end{tikzpicture}
     \qquad
     \begin{tikzpicture}[scale=1.5]
      \node[invisivertex] (O) at (2.5,2){\tiny $000$};

      \node[invisivertex] (H1+) at (0,1){\tiny $0++$};
      \node[invisivertex] (H2+) at (1,1){\tiny $+0+$};
      \node[invisivertex] (H3-) at (2,1){\tiny $-+0$};
      \node[invisivertex] (H2-) at (3,1){\tiny $-0-$};
      \node[invisivertex] (H1-) at (4,1){\tiny $0--$};
      \node[invisivertex] (H3+) at (5,1){\tiny $+-0$};

      \node[invisivertex] (C1) at (0,0){\tiny $+++$};
      \node[invisivertex] (C2) at (1,0){\tiny $-++$};
      \node[invisivertex] (C3) at (2,0){\tiny $-+-$};
      \node[invisivertex] (C4) at (3,0){\tiny $---$};
      \node[invisivertex] (C5) at (4,0){\tiny $+--$};
      \node[invisivertex] (C6) at (5,0){\tiny $+-+$};

      \node[invisivertex] (1) at (2.5,-1){\tiny $\hat 0$};

      \path[-] (O) edge [] node[above] {} (H1+);
      \path[-] (O) edge [] node[above] {} (H2+);
      \path[-] (O) edge [] node[above] {} (H3-);
      \path[-] (O) edge [] node[above] {} (H2-);
      \path[-] (O) edge [] node[above] {} (H1-);
      \path[-] (O) edge [] node[above] {} (H3+);

      \path[-] (C1) edge [] node[above] {} (H1+);
      \path[-] (C1) edge [] node[above] {} (H2+);
      \path[-] (C2) edge [] node[above] {} (H1+);
      \path[-] (C2) edge [] node[above] {} (H3-);
      \path[-] (C3) edge [] node[above] {} (H3-);
      \path[-] (C3) edge [] node[above] {} (H2-);
      \path[-] (C4) edge [] node[above] {} (H2-);
      \path[-] (C4) edge [] node[above] {} (H1-);
      \path[-] (C5) edge [] node[above] {} (H1-);
      \path[-] (C5) edge [] node[above] {} (H3+);
      \path[-] (C6) edge [] node[above] {} (H3+);
      \path[-] (C6) edge [] node[above] {} (H2+);

      \path[-] (C1) edge [dashed] node[above] {} (1);
      \path[-] (C2) edge [dashed] node[above] {} (1);
      \path[-] (C3) edge [dashed] node[above] {} (1);
      \path[-] (C4) edge [dashed] node[above] {} (1);
      \path[-] (C5) edge [dashed] node[above] {} (1);
      \path[-] (C6) edge [dashed] node[above] {} (1);
      
      \end{tikzpicture}
  \end{center}
  The $\aaa\bbb$-index of $\Sigma\cup\{\hat 0\}$ can be computed as
  \[
    \aaa^3 + 5\aaa\bbb\aaa + 5 \aaa^2\bbb + \aaa\bbb^2 = \aaa(\aaa^2 + 5\bbb\aaa + 5\aaa\bbb + \bbb^2) = \aaa\cdot\pullPsi(\cL;\aaa,\bbb)\,.
  \]
  % As seen above, this equals $\aaa\cdot\pullPsi(\cL;\aaa,\bbb)$.
\end{example}

\subsection{Main results}
\label{sec:mainresults}

The main results of this paper concern \emph{$R$-labeled posets}.
These form a large family of posets including \emph{distributive lattices}, \emph{geometric lattices}, and \emph{semimodular lattices}.
In order to state \Cref{thm:combinatorial-interp}, we introduce a combinatorial statistic on maximal chains of these posets and use this to describe the extended $\aaa\bbb$-index.
In \Cref{sec:connections}, we briefly discuss this combinatorial statistic for general edge labeled graded posets.

A function~$\lambda$ from the set of cover relations $X\lessdot Y$ in~$P$ into the positive integers is an \Dfn{$R$-labeling} of $P$ if, for every interval $[X, Y]$ in~$P$, there is a unique
maximal chain $X=\cM_i\lessdot \cM_{i+1} \lessdot \cdots \lessdot \cM_j=Y$ such that 
\[ 
  \lambda(\cM_i,\cM_{i+1}) \leq \lambda(\cM_{i+1},\cM_{i+2}) \leq \cdots \leq \lambda(\cM_{j-1},\cM_j). 
\] 
We say a poset $P$ is \Dfn{$R$-labeled} if it is finite, graded, and admits an
$R$-labeling.
Throughout this section, we consider $R$-labeled posets with a fixed $R$-labeling~$\lambda$.
%Many interesting families of posets admit
%$R$-labelings. Most importantly for the present context, geometric lattices,
%\ie, lattices of flats for finite matroids, admit $R$-labelings,
%see~\cite[Example 3.8]{bjorner-shellings}. 

The first result is
a combinatorial statistic describing the coefficients of the extended
$\aaa\bbb$-index which witnesses their nonnegativity.
It generalizes~\cite[Corollary~7.2]{billera-ehrenborg-readdy} and also reproves it using purely combinatorial arguments.
For a maximal chain~$\cM =
\{\cM_0\lessdot\cM_1\lessdot\dots\lessdot\cM_n\}$ in $P$, define the monomial
$\StatMon(\cM) = u_1\cdots u_n$ in $\aaa,\bbb$ given by $u_1 = \aaa$
and for $i \in \{2,\dots,n\}$ by
\begin{equation}
\label{eq:statmon}
  u_i = \begin{cases}
                \aaa &\text{if } \lambda(\cM_{i-2},\cM_{i-1}) \leq \lambda(\cM_{i-1},\cM_{i})\,, \\
                \bbb &\text{if } \lambda(\cM_{i-2},\cM_{i-1}) > \lambda(\cM_{i-1},\cM_{i})\,.
              \end{cases}
\end{equation}
Now, let $E \subseteq \{1,\dots,n\}$, viewed as a subset of the cover relations
in the chain~$\cM$. Define the monomial $\StatMon(\cM,E) = v_1\dots v_n$ in
$\aaa,\bbb$ to be obtained from $\StatMon(\cM)$ by
\begin{itemize}
  \item replacing all variables~$\aaa$ by $\bbb$ at positions~$i\in \{1,\dots,
  n\}$ if $i \in E$ and
  \item replacing all variables~$\bbb$ by $\aaa$ at positions~$i\in \{2,\dots,
  n\}$ if $i-1 \in E$.
\end{itemize}
% In symbols this means, for the given position~$i \in \{1,\dots,n\}$, that
% \begin{equation*}
%   \begin{array}{ll}
%           v_i = \aaa &\text{ if } \begin{cases}
%                               u_i = \aaa, \quad i\hspace*{21.5pt} \notin E\quad\text{or} \\
%                               u_i = \bbb, \quad i-1 \in E\,,
%                             \end{cases} \\ \\
%           v_i = \bbb & \text{ if } \begin{cases}
%                               u_i = \aaa, \quad i\hspace*{21.5pt} \in E\quad\text{or} \\
%                               u_i = \bbb, \quad i-1 \notin E\,.
%                             \end{cases}
%   \end{array}
% \end{equation*}
In particular, we have $\StatMon(\cM,\emptyset) = \StatMon(\cM)$, and $v_1=\bbb$ if and only if $1 \in E$.
% \begin{equation}
% \label{eq:firstindex}
%   v_1 = \begin{cases}
%           \aaa & \text{ if } 1 \notin E\,, \\
%           \bbb & \text{ if } 1 \in E\,.
%         \end{cases}
% \end{equation}

\begin{thm}
\label{thm:combinatorial-interp}
  Let~$P$ be an $R$-labeled poset of rank~$n$.
 Then 
  \[
    \extPsi(P;y,\aaa,\bbb)
      = \sum_{(\cM,E)} y^{\#E}\cdot\StatMon(\cM,E)
  \]
  where the sum ranges over all maximal chains~$\cM$ in~$P$ and all subsets~$E \subseteq \{1,\dots,n\}$.
\end{thm}

When $P$ is a geometric lattice, setting $y = 0$ in \Cref{thm:combinatorial-interp} recovers~\cite[Corollary~7.2]{billera-ehrenborg-readdy}.
Specifically
% The special case for the $\aaa\bbb$-index is then~\cite[Corollary~7.2]{billera-ehrenborg-readdy} given by
$\Psi(P;\aaa,\bbb) = \sum_{\cM} \StatMon(\cM)$,
where the sum ranges over all maximal chains~$\cM = \{ \cM_0 \lessdot \dots \lessdot \cM_n\}$.

\begin{example}
\label{ex:face-poset3}
  The poset from the previous examples admits the $R$-labeling given below on the left.
  On the right, we collect the relevant data to compute the combinatorial description of the extended $\aaa\bbb$-index.
  \begin{center}
    \begin{tikzpicture}[scale=.7, baseline={([yshift=-.5ex]current bounding box.center)}]
      \node[invisivertex] (1) at (2,4){$\hat 1$};
      \node[invisivertex] (H1) at (0,2){$\alpha_1$};
      \node[invisivertex] (H2) at (2,2){$\alpha_2$};
      \node[invisivertex] (H3) at (4,2){$\alpha_3$};
      \node[invisivertex] (0) at (2,0){$\hat 0$};

      \node[edgelabel] at (.5,1){\tiny $1$};
      \node[edgelabel] at (1.7,1){\tiny $2$};
      \node[edgelabel] at (3.5,1){\tiny $3$};

      \node[edgelabel] at (.5,3){\tiny $2$};
      \node[edgelabel] at (1.7,3){\tiny $1$};
      \node[edgelabel] at (3.5,3){\tiny $1$};

      \draw (0) -- (H1) -- (1);
      \draw (0) -- (H2) -- (1);
      \draw (0) -- (H3) -- (1);
    \end{tikzpicture}
    \hspace*{2cm}
    \begin{tabular}{|c|c|c|c|c|}
    \hline
    & & & &\\[-10pt]
    $E$ & $y^{\#E}$
    & $\hat 0 \lessdot \alpha_1 \lessdot \hat 1$
    & $\hat 0 \lessdot \alpha_2 \lessdot \hat 1$ &
    $\hat 0 \lessdot \alpha_3 \lessdot \hat 1$ \\[5pt]
    \hline
    & & & &\\[-10pt]
    $\{\}$      & $1$   & $\aaa\aaa$ & $\aaa\bbb$ & $\aaa\bbb$ \\
    $\{ 1 \}$   & $y$   & $\bbb\aaa$ & $\bbb\aaa$ & $\bbb\aaa$ \\
    $\{ 2 \}$   & $y$   & $\aaa\bbb$ & $\aaa\bbb$ & $\aaa\bbb$ \\
    $\{ 1,2 \}$ & $y^2$ & $\bbb\bbb$ & $\bbb\aaa$ & $\bbb\aaa$\\[5pt]
    \hline
    \end{tabular}
  \end{center}
  Then $\extPsi(\cL;y,\aaa,\bbb) = \aaa\aaa + (3y+2y^2)\bbb\aaa + (2 + 3y)\aaa\bbb + y^2\bbb\bbb$.
\end{example}

% \Cref{thm:combinatorial-interp} has many consequences, which we formulate in eight corollaries.
% The most important gives a substitution sending the
% $\aaa\bbb$-index to the extended $\aaa\bbb$-index---meaning that the extended
% $\aaa\bbb$-index is already encoded in the $\aaa\bbb$-index.

\begin{corollary}
\label{thm:refinement-of-ber}
  For an $R$-labeled poset~$P$, we have
  \[
    \extPsi(P;y,\aaa,\bbb) = \omega\big(\Psi(P;\aaa,\bbb)\big)
  \]
  where the substitution $\omega$ replaces all occurrences of $\aaa\bbb$ with
  $\aaa\bbb + y\bbb\aaa + y\aaa\bbb + y^2\bbb\aaa$ and then simultaneously
  replaces all remaining occurrences of $\aaa$ with $\aaa+y\bbb$ and $\bbb$ with
  $\bbb+y\aaa$.
\end{corollary}

% \begin{example}
%   We have seen that the poset~$\cL$ from \Cref{ex:face-poset2} has $\aaa\bbb$-index
%   \[
%     \Psi(\cL;\aaa,\bbb) = \extPsi(\cL,0,\aaa,\bbb) = \aaa\aaa + 2\aaa\bbb\,.
%   \]
%   Applying $\omega$ gives
%   \[
%     \omega\big(\Psi(\cL;\aaa,\bbb)\big)
%       = (\aaa+y\bbb)^2 + 2(\aaa\bbb + y\bbb\aaa + y\aaa\bbb + y^2\bbb\aaa) = \extPsi(\cL;y,\aaa,\bbb)\,,
%   \]
%   which coincides with the extended $\aaa\bbb$-index we computed in \Cref{ex:face-poset3}.
% \end{example}

Using \Cref{thm:refinement-of-ber}, the monomials $\StatMon(\cM,E)$ in \Cref{thm:combinatorial-interp} capture the same information as the \emph{generalized descent sets} on \emph{réseaux} as defined by Bergeron, Mykytiuk, Sottile, and van Willigenburg in~\cite[Section~7]{bergeron-mykytiuk-sottile-vanWilligenburg2000} in the context of quasisymmetric functions.
The next corollary can be seen as a refinement of~\cite[Proposition~2.2]{stembridge-p-partitions} and of~\cite[Theorem~7.2]{bergeron-mykytiuk-sottile-vanWilligenburg2000}, stated in terms of $\aaa\bbb$-indices rather than quasisymmetric functions.
Both can be seen as the special case for the pullback $\aaa\bbb$-index: the first for \emph{enriched $P$-partitions} and the second for general edge-labeled graded posets, compare with \Cref{sec:connections}.
We start by describing their relevant combinatorics in the present notation.
Let $\cM$ be a maximal chain with $\StatMon(\cM) = u_1\dots u_n$, and let
\[
  \peak(\cM) = \big\{ i \in \{2,\dots,n\} \mid u_{i-1} = \aaa, u_i = \bbb \big\}
\]
denote its \Dfn{peak set}.
A set $S \subseteq \{1,\dots,n\}$ is then \Dfn{$\cM$-peak-covering} if
\[
  \peak(\cM) \subseteq S \cup \{ i+1 \mid i \in S\}\,.
\]
For $\StatMon(\cM, S) = v_1\cdots v_n$, let $\bout(\cM,S)$ be the number of positions $i \in \{1,\dots,n\} \setminus S$ where $v_i = \bbb$.

\begin{corollary}
\label{cor:peakenumeration}
  For an~$R$-labeled poset $P$ of rank~$n$, we have
  \[
    \extPsi(P;y,\aaa,\bbb) = \sum_{(\cM,S)} (1+y)^{\# S}\cdot y^{\bout(\cM,S)} \cdot \wt_S\,,
  \]
  where the sum ranges over all maximal chains~$\cM$ and all $\cM$-peak-covering subsets $S \subseteq \{1,\dots,n\}$ and where $\wt_S = w_1\dots w_n$ as given in \Cref{eqn:ext-weights}.
\end{corollary}

% \begin{example}
%   For the poset~$\cL$ from \Cref{ex:face-poset2}, using \Cref{cor:peakenumeration} we compute
%   \begin{align*}
%     \extPsi(\cL;y,\aaa,\bbb)
%     &= \underbrace{(\aaa-\bbb)^2}_{S = \emptyset} + \underbrace{(1+y) \cdot \bbb(\aaa-\bbb)}_{S = \{1\}} + \underbrace{(1+y) \cdot (\aaa-\bbb)\bbb}_{S = \{2\}} + \underbrace{(1+y)^2 \cdot\bbb^2}_{S = \{1,2\}} \\
%     &+ 2\cdot\Big(\underbrace{(1+y) \cdot y \cdot \bbb(\aaa-\bbb)}_{S = \{1\}} + \underbrace{(1+y) \cdot (\aaa-\bbb)\bbb}_{S = \{2\}} + \underbrace{(1+y)^2 \cdot\bbb^2}_{S = \{1,2\}} \Big)
%   \end{align*}
%   where the sum in the first row corresponds to the maximal chain $\cM = \{ \hat 0 \lessdot \alpha_1 \lessdot \hat 1\}$ with $\StatMon(\cM) = \aaa\aaa$ and $\peak(\cM) = \emptyset$.
%   The condition of being $\cM$-peak-covering is thus vacuous, and we sum over all $S \subseteq \{1,2\}$.
%   The sum in the second row corresponds to the two maximal chains $\cM = \{ \hat 0 \lessdot \alpha_i \lessdot \hat 1\}$ for $i \in \{2,3\}$ with $\StatMon(\cM) = \aaa\bbb$ and $\peak(\cM) = \{2\}$.
%   In this case, the condition of being $\cM$-peak-covering excludes the empty set, and we sum over all nonempty subsets $S \subseteq \{1,2\}$.
%   Observe that only in the second row, we have the second letter of $\StatMon(\cM)$ equal to~$\bbb$, so the only set for which $\bout(\cM,S) \neq 0$ is $S = \{1\}$, and in this case $\bout(\cM,S) = 1$.
%   The above sum further expands to
%   \begin{align*}
%     \extPsi(\cL;y,\aaa,\bbb)
%     &= (\aaa\aaa + y\bbb\aaa + y\aaa\bbb + y^2\bbb\bbb)\ +\ 2\big((y+y^2)\bbb\aaa + (1+y)\aaa\bbb \big) \\
%     &= \aaa\aaa + (3y+2y^2)\bbb\aaa + (2+3y)\aaa\bbb + \bbb\bbb\,.
%   \end{align*}
% \end{example}

Another consequence of \Cref{thm:refinement-of-ber} is that the Poincaré polynomial
of~$P$ is in fact encoded in its $\aaa\bbb$-index. To see this, we define another
substitution~$\iota$, which deletes the first letter from every $\aaa\bbb$-monomial, so
$\iota(\aaa^3\bbb\aaa + (1+y)\bbb\aaa) = \aaa^2\bbb\aaa + (1+y)\aaa$ for example. 
This gives us a way to obtain the Poincaré polynomial from the $\aaa\bbb$-index, a result which is similar in spirit to~\cite[Proposition 5.3]{billera-ehrenborg-readdy}.

\begin{corollary}\label{eq:exabpoin}
  For an $R$-labeled poset $P$ of rank $n$, the Poincaré polynomial is the coeffcient of $\aaa^{n-1}$ in
  $\iota\big(\omega\big(\Psi(P;\aaa,\bbb)\big)\big)$.
\end{corollary}

\Cref{thm:refinement-of-ber}
generalizes~\cite[Theorem~3.1]{billera-ehrenborg-readdy} relating the
$\aaa\bbb$-index of the lattice of flats of an oriented matroid with the
$\aaa\bbb$-index of its face poset. As a consequence, we see that
$\extPsi(P;y,\aaa,\bbb)$ is akin to a refinement of a $\ccc\ddd$-index. We
make this observation precise in the following corollary. 

\begin{corollary}
\label{cor:cd-index}
  For an $R$-labeled poset~$P$, there exists a polynomial $\Phi(P;
  \ccc_1,\ccc_2,\ddd)$ in noncommuting variables $\ccc_1,\ccc_2,\ddd$ such that 
  \[ 
    \extPsi(P; y, \aaa, \bbb) = \Phi(P;\, \aaa+y\bbb,\, \bbb+y\aaa,\, \aaa \bbb + y \bbb \aaa + y \aaa \bbb + y^2 \bbb \aaa) .
  \]
  In particular, the pullback $\aaa\bbb$-index $\pullPsi(P;\aaa,\bbb)$ is a polynomial in noncommuting variables $\ccc = \aaa+\bbb$ and $2\ddd = 2(\aaa\bbb+\bbb\aaa)$.
\end{corollary}

\begin{remark}[The synthetic $\ccc\ddd$-index]\label{rem:matroids}
  Recall that the $\ccc\ddd$-index of a poset exists if the $\aaa\bbb$-index can be written as a polynomial in $\ccc = \aaa + \bbb$ and $\ddd = (\aaa\bbb + \bbb\aaa)$.
  Bayer and Klapper proved a conjecture of Fine that a poset satisfies the \emph{generalized Dehn-Sommerville relations} if and only if its $\ccc\ddd$-index exists and has integer coefficients~\cite[Theorem 4]{bayer-klapper}.
  % Motivated  core questions related to the $\ccc\ddd$-index of a poset are: (a) when does it exist? and (b) when are the coefficients nonnegative?
  The $\ccc\ddd$-index of an Eulerian poset always exists (see~\cite[Theorem 2.1]{bayer-billera}) and has nonnegative coefficients when it comes from the face poset of a shellable regular $CW$ sphere like the face poset of a convex polytope~\cite[Theorem 2.2]{stanley-flag-f} (or, more generally, from a Gorenstein* poset~\cite[Theorem 1.3]{karu}).
  % If the poset is the face poset of a polytope or---more generally---Goreinstein*, then the coefficients of the $\ccc\ddd$-index are nonnegative~\cite[Theorem 1.3]{karu}.
  
  In~\cite{billera-ehrenborg-readdy}, Billera, Ehrenborg, and Readdy give an elegant alternative proof of the nonnegativity of the $\ccc\ddd$-index of the face poset of an oriented matroid.
  They use the support map from \Cref{eq:supppre} to relate the $\aaa\bbb$-index of the lattice of flats to the $\aaa\bbb$-index of the face poset.
  In our language, they interpret (using posets and polytopes) the extended $\aaa\bbb$-index of an oriented matroid at $y=0$ and $y=1$.
  % They viewed these as two different polynomials, connected by poset statements.
  % The extended $\aaa\bbb$-index is a single polynomial interpolating between their two polynomials.
  % As we will see in \Cref{rem:oig-version,,rem:jp-version}, the extended $\aaa\bbb$-index unifies this with several analogous results from the contexts of oriented interval greedoids and distributive lattices.
  Every matroid admits an extended $\aaa\bbb$-index, and the evaluation at $y=0$ is the $\aaa\bbb$-index of its lattice of flats.
  This raises the natural question whether there is a geometric or poset-theoretic interpretation of the $y=1$ evaluation of the extended $\aaa\bbb$-index.
  For this reason, we call the $y=1$ evaluation of the extended $\aaa\bbb$-index rewritten in terms of $\ccc$ and $\ddd$ the \Dfn{synthetic $\ccc\ddd$-index}.
  % It would be too much to ask for a polytope whose face lattice has the correct $\ccc\ddd$-index (we illustrate this for two different matroids, below), but there may be a more general geometric construction capturing this structure.
\end{remark}

  \begin{example}[The Fano matroid]
  % As a first example, the Fano matroid to illustrate some of the subtlety of the previous question.
  Setting $y=1$ and then $\ccc = \aaa + \bbb$ and $\ddd = \aaa\bbb + \bbb\aaa$ in the extended $\aaa\bbb$-index of the \emph{Fano matroid}~\cite[Example~6.6.2(1)]{BjornerEtAl} gives the synthetic $\ccc\ddd$-index of the Fano matroid: $12\ccc\ddd + 28\ddd\ccc + \ccc^3$.
  A convex $3$-polytope with this $\ccc\ddd$-index would have $30$~vertices and $14$~facets; see~\cite{Meisinger}.
 Thus its polar dual polytope would have $14$~vertices and $30$~facets, contradicting the the Upper Bound Theorem~\cite[p.180]{McMullen}.
\end{example}

 \begin{example}[The Mac Lane matroid]
 We compute the synthetic $\ccc\ddd$-index of the \emph{Mac Lane matroid}; see~\cite[page~114]{bland-lasvergnas} and~\cite[Section~2]{ziegler-nonorientable}.
  % \josh{Just throwing this question out there: do we want to mention Mac Lane here since he originally defined $G_3$?}
  We get the synthetic $\ccc\ddd$-index $18\ccc\ddd + 32\ddd\ccc + \ccc^3$, which is the $\ccc\ddd$-index of a polytope!
\end{example}

\begin{remark}[Oriented interval greedoids]
\label{rem:oig-version}
  The argument used for oriented matroids and their lattices of flats also applies to \emph{oriented interval greedoids}, where the analogue of \Cref{eq:supppre} is given in~\cite[Theorem~6.8]{saliola-thomas}.
  Since the lattice of flats of an interval greedoid is a semimodular lattice%
  %~\cite[Theorem~8.8.7]{white}
  , it admits an $R$-labeling; see~\cite[Theorem~3.7]{bjorner-shellings}.
  Applying \Cref{thm:refinement-of-ber} and setting $y=1$ gives~\cite[Corollary~6.12]{saliola-thomas}.
\end{remark}

\begin{remark}[Distributive lattices \& $r$-signed Birkhoff posets]
\label{rem:jp-version}
  Ehrenborg discussed an $\omega$-like substitution for arbitrary distributive lattices~\cite{Ehrenborg-signed}.
  Remarkably, that substitution is equivalent to the substitution in \Cref{thm:refinement-of-ber} for $y = r-1 \in \N$.
  In that case of distributive lattices, the parameter~$r$ is a fixed integer (rather than a variable) carrying information about the fiber sizes of a certain support map.
  For a (not necessarily graded) finite poset~$P$, the \emph{$r$-signed Birkhoff poset} $J_r(P)$ is the collection of pairs $(F,f)$ where~$F$ is an \emph{order ideal} in~$P$ and~$f$ is a map from the maximal elements in~$F$ to the set $\{1,\dots,r\}$, with order relation given by
  \[
    (F,f) \leq (G,g) \quad \Longleftrightarrow \quad G\subseteq F \text{ and } f(x) = g(x) \text{ for all }x \in \max(F) \cap \max(G)\,.
  \]
  These posets were defined in~\cite{Hsiao,Ehrenborg-signed} and studied in connection to the Birkhoff lattice $J(P) = J_1(P)$.
  The map $z: J_r(P) \to J(P)$ with $(F,f) \mapsto F$ is an order- and rank-preserving poset surjection for which the fiber size of a chain~$\cC$ in $J(P)$ can---in the notation from the previous sections---be computed by
  \(
    \# z^{-1}(\cC) = \Poin_{\cC}(J(P);r-1),
  \)
  see ~\cite[Proposition 5.2]{Ehrenborg-signed}.
  Since distributive lattices are modular, they admit $R$-labelings; see~\cite[Theorem~3.7]{bjorner-shellings}.
  Thus, applying~\Cref{thm:refinement-of-ber} for $y=r-1$ gives the first part of~\cite[Theorem 4.2]{Ehrenborg-signed}.
\end{remark}

% \medskip
% By examining the statistic from~\Cref{thm:combinatorial-interp}, we see that the
% extended $\aaa\bbb$-index has symmetry among its coefficients. We encode this in
% the following theorem, which generalizes the bivariate version
% of~\cite[Theorem~A]{Maglione-Voll}. 

% \begin{corollary}
% \label{thm:symmetry-of-coeffs}
%   Let~$P$ be an $R$-labeled poset of rank~$n$.
%   Let moreover~$\cM$ be a maximal chain in~$P$ and let $E, E^c$ be complementary subsets of $\{1,\dots,n\} = E \cup E^c$.
%   Then $\StatMon(\cM,E) = u_1\dots u_{n}$ and $\StatMon(\cM,E^c) = u_1^c\dots u_{n}^c$ are complementary words, \ie, $\{u_i,u_i^c\} = \{\aaa,\bbb\}$ for every $i \in \{1,\dots,n\}$.
%   In particular, the coefficients of $y^\ell\mon$ and of $y^{n-\ell}\mon^c$ in $\extPsi(P;q,\aaa,\bbb)$ coincide,
%   \[
%     [y^\ell \mon]\ \extPsi(P;y,\aaa,\bbb) = [y^{n-\ell} \mon^c]\ \extPsi(P;y,\aaa,\bbb)\,,
%   \]
%   for complementary words~$\mon$ and~$\mon^c$ in $\aaa,\bbb$ and for $\ell\in
%   \{0,\dots, n\}$.
% \end{corollary}

We next turn toward the coarse flag Hilbert--Poincaré series introduced and
studied in~\cite{Maglione-Voll}. The numerator of this rational function is
defined in~\cite[Equation~(1.13)]{Maglione-Voll}, and we extend this definition
to graded posets via
\[
  \Num(P;y,t) = \sum_{\cC\text{ chain in } P\setminus\{\hat{0},\hat 1\}} \POIN[\{\hat 0\} \cup \cC]{P}{y}\cdot t^{\#\cC}(1-t)^{n-1-\#\cC}\, \in \Z[y,t]\,.
\]

% \begin{example}
%   Let~$\cL$ be the poset from \Cref{ex:face-poset2}.
%   This poset has rank~$2$ and numerator polynomial
%   \[
%     \Num(P;y,t) = (1 + 3y + 2y^2)(1-t) + 3\cdot(1+y)^2t = 1 + 3y + 2y^2 + (2 + 3y + y^2)t\,.
%   \]
% \end{example}

% The following corollary generalizes the property in \Cref{eq:ab01} to extended
% $\aaa\bbb$-indices and relates them to coarse flag Hilbert--Poincaré series.
% Recall from \Cref{eq:exabpoin}, the function $\iota$ that deletes the first
% letter from every $\aaa\bbb$-monomial.

% \begin{corollary}
% \label{cor:enab0}
%   For an $R$-labeled poset~$P$, we have
%   \[
%     \iota\big(\extPsi(P;y,\aaa,\bbb)\big) = \sum_{\cC\text{ chain in } P\setminus\{\hat 0,\hat 1\}} \Poin_{\{\hat 0\} \cup \cC}(P;y) \cdot \wtp_\cC\ \in \N[y]\langle \aaa,\bbb\rangle \,,
%   \]
%   where $\wtp_\cC = w_1\dots w_{n-1}$.
% \end{corollary}

By removing the first letter of every $\aaa\bbb$ monomial and then specializing via $\aaa \mapsto 1$ and $\bbb \mapsto t$
we obtain a proof of~\cite[Conjecture~E]{Maglione-Voll} and its generalization to $R$-labeled posets:


\begin{corollary}
\label{thm:maglione-voll-conjecture}
  For an $R$-labeled poset~$P$, the coefficients of $\Num(P;y,t)$ are
  nonnegative.
\end{corollary}

Together with \Cref{eq:exabpoin}, we obtain $\Poin(P;y) = [t^0]\ \Num(P;y,t)$.
The substitutions in the previous corollaries show that \Cref{thm:combinatorial-interp} also gives analogous combinatorial interpretations for the coefficients of  $\iota\big(\extPsi(P;y,\aaa,\bbb)\big)$ and of $\Num(P;y,t)$. 

% \begin{example}
%   Recall that $\iota$ is the function that deletes the first letter of every
%   $\aaa\bbb$-monomial. Thus, ignoring the first position in the computation in
%   \Cref{ex:face-poset3} yields
%   \[
%     \iota\big(\extPsi(\cL;y,\aaa,\bbb)\big) = (1 + 3y + 2y^2)\aaa + (2 + 3y + y^2)\bbb\,.
%   \]
%   The substitution $\aaa \mapsto 1$ and $\bbb \mapsto t$ yields $\Num(\cL;y,t)$.
%   Since $n=2$, we have
%   \[
%     [\aaa^{n-1}]\ \iota\big(\extPsi(\cL;y,\aaa,\bbb)\big) = [t^0]\ \Num(\cL;y,t) = \Poin(\cL;y)\,,
%   \]
%   also in agreement with \Cref{eq:exabpoin}.
% \end{example}

\begin{remark}[Geometric semilattices]
  Note that~\cite[Conjecture~E]{Maglione-Voll} concerns all hyperplane arrangements (central and affine).
While the intersection posets of central hyperplane arrangements are geometric lattices and, thus, admit $R$-labelings~\cite[Example 3.8]{bjorner-shellings}, the intersection posets of affine arrangements are part of a more general family called \emph{geometric semilattices}, first explicitly studied by Wachs and Walker in~\cite{wachs-walker}.
A theorem of Ziegler shows that if $\cL$ is a geometric semilattice, then $\cL\cup\{\hat{1}\}$ admits an $R$-labeling~\cite[Theorem 2.2]{ziegler-shellability}.
Thus~\Cref{thm:combinatorial-interp} holds for intersection posets of affine arrangements. %(now with a formal unique maximal element included, which is consistent with the formulation in~\cite[Conjecture~E]{Maglione-Voll}).
\end{remark}

\begin{remark}[Implications for other zeta functions]\label{rem:implications-for-zetas}
The coarse flag Hilbert--Poincaré polynomial of a poset $P$ comes from a natural
specialization of its flag Hilbert--Poincaré series. The flag Hilbert--Poincaré
series is a rational function in $\mathbb{Q}[y](t_x \mid x \in P)$ given by 
\begin{align*}
  \mathsf{fHP}_P(y, \mathbf{t}) &= \sum_{\cC \text{ chain in } P\setminus \hat{0}} \POIN[\cC]{P}{y} \prod_{x\in \cC} \dfrac{t_x}{1-t_x}\,.
\end{align*}
The coarse flag Hilbert--Poincaré polynomial $\Num(P; y, t)$ is obtained by setting all the $t_x$ equal to $t$ and considering $(1 - t)^{\rank(P)}\mathsf{fHP}_P(y,t)$. 
Different specializations of $\mathsf{fHP}_P(y, \mathbf{t})$ yield other well-studied zeta functions like local Igusa zeta functions of hyperplane arrangements~\cite{Budur-Nero-Mustata}, motivic zeta functions of matroids from~\cite{Jensen-Kutler-Usatine}, and the conjugacy class counting zeta functions of certain group schemes defined in~\cite{Rossmann-Voll}.
Moreover, each of these is obtained from $\mathsf{fHP}_P(y, \mathbf{t})$ by a monomial substitution of the form $y = -p^{-1}$ and $t_x = p^{\lambda_x}t^{\mu_x}$ for some integers $\lambda_x$ and $\mu_x$, where $p$ is a prime and $t=p^{-s}$ for a complex variable~$s$; see \cite[Remark 1.3]{Maglione-Voll}.
% In~\cite{Maglione-Voll}, the flag Hilbert--Poincaré series is obtained from an evaluation of a zeta function.
% In each of these settings, the poles and their orders play an important role in understanding how quickly the terms of these zeta functions grow.
% Poles of zeta functions are challenging to compute in general---see for example \cite[Conjectures~2.3.1 \& 2.3.2]{Denef}, \cite[Conjecture~IV]{Rossmann}, and \cite[Question~1.8]{Rossmann-Voll}.
% \Cref{thm:maglione-voll-conjecture} tells us that $\mathsf{fHP}_P(y,t)$ has exactly one pole at $t=1$ of order $\rank(P)$.
% In particular, this result suggests there might be similar combinatorial interpretations of the numerators of zeta functions obtained from specializing $\mathsf{fHP}_P(y, \mathbf{t})$. 
% For example, the local Igusa zeta functions of hyperplane arrangements~\cite{Budur-Nero-Mustata}, the motivic zeta functions of matroids from~\cite{Jensen-Kutler-Usatine}, and the conjugacy class counting zeta functions
% of certain group schemes defined in~\cite{Rossmann-Voll} are all obtained from $\mathsf{fHP}_P(y; \mathbf{t})$ by monomial substitutions of the form $y = -p^{-1}$ and $t_x = p^{\lambda_x}t^{\mu_x}$ for some integers $\lambda_x$ and $\mu_x$, where $p$
% is a prime and $t=p^{-s}$ for a complex variable $s$; see \cite{Maglione-Voll}.

% This result suggests that other interesting specializations of $\mathsf{fHP}_P(y; \mathbf{t})$ could have combinatorial descriptions, which can be used to understand the poles of $\mathsf{fHP}_P(y; \mathbf{t})$.
% Although the substitution to obtain $\Num(P; y, t)$ is
% comparatively simpler than the other ones mentioned,
\end{remark}

The specialization of $\Num(P;y,t)$ at $y=1$ was studied further for matroids
and oriented matroids by the second author and Kühne in~\cite{Kuhne-Maglione},
who showed $\Num(P;1,t)$ is the sum of $h$-polynomials of simplicial complexes
related to the chambers if $P$ is the lattice of flats of a real central
hyperplane arrangement. The following corollary proves a generalized version of the conjectured lower
bound from~\cite[Conjecture~1.4]{Kuhne-Maglione}.

\begin{corollary}
\label{thm:lower-bound-on-coefficients}
  Let~$P$ be an $R$-labeled poset of rank~$n$.
  The coefficient of $t^k$ in $\Num(P;1,t)$ is bounded below by
   \(
     \binom{n-1}{k}\cdot \POIN{P}{1}.
   \)
\end{corollary}

% \begin{Beyond $R$-labelings}
% \label{sec:beyondR}

% We have presented all results for $R$-labeled posets.
% In that setting, \Cref{thm:combinatorial-interp} shows that the extended $\aaa\bbb$-index has nonnegative coefficients.
% Nonnegativity may fail, however, for posets that do not admit $R$-labelings.
% For example, the graded poset
% \begin{center}
%   \begin{tikzpicture}[scale=.7]
%     \node[fill=black, circle, inner sep=2pt]  (0) at  (0,0) {};
%     \node[fill=black, circle, inner sep=2pt] (X1) at (-1,1) {};
%     \node[fill=black, circle, inner sep=2pt] (X2) at ( 1,1) {};
%     \node[fill=black, circle, inner sep=2pt] (Y1) at (-1,2) {};
%     \node[fill=black, circle, inner sep=2pt] (Y2) at ( 1,2) {};
%     \node[fill=black, circle, inner sep=2pt]  (1) at  (0,3) {};
%     \draw (0) -- (X1) -- (Y1) -- (1);
%     \draw (0) -- (X2) -- (Y2) -- (1);
%     % \node[edgelabel] at ( .7, .3){\tiny $1$};
%     % \node[edgelabel] at (1.3,1.5){\tiny $2$};
%     % \node[edgelabel] at ( .7,2.7){\tiny $1$};
%     % \node[edgelabel] at (- .7, .3){\tiny $2$};
%     % \node[edgelabel] at (-1.3,1.5){\tiny $1$};
%     % \node[edgelabel] at (- .7,2.7){\tiny $2$};
%   \end{tikzpicture}
% \end{center}
% does \emph{not} admit an $R$-labeling and its extended $\aaa\bbb$-index
% \begin{align*}
%   \aaa\aaa\aaa &+ (-1+2y)\aaa\aaa\bbb + (1+2y)\aaa\aaa\bbb + y(2 + y^2)\bbb\aaa\aaa + (2y^2 - 1)\aaa\bbb\bbb \\
%   &+ (-y^3 + 2y^2) \bbb\aaa\bbb + y^2(2 + y)\bbb\bbb\aaa + y(3y^2 + 2y - 2)\bbb\bbb\bbb
% \end{align*}
% has some \emph{negative} coefficients.
% % In particular, this means that the coefficients of the Poincaré-extended $\aaa\bbb$-index are not nonnegative for all graded posets.
% One may still define a manifestly-positive (\Dfn{combinatorial}) \Dfn{extended $\aaa\bbb$-index} of a \emph{general} finite edge-labeled graded poset of rank~$n$ by
% \[
%   \combextPsi(P;y,\aaa,\bbb) = \sum_{(\cM,E)} y^{\#E}\cdot\StatMon(\cM,E) \ \in \N[y]\langle\aaa,\bbb\rangle\,.
% \]
% While this combinatorial description may not always be linked to the Poincaré polynomial, the proofs of \Cref{thm:refinement-of-ber,,cor:peakenumeration} still hold.
% we \Cref{thm:refinement-of-ber,,cor:peakenumeration} by proving the statement for every maximal chain~$\cM$ (\Cref{eq:extending-equality,,eq:peakenumeration}), and then summing over all maximal chains.
% This suggests that $R$-labeled posets are a (rather general) family of posets for which the Poincaré-extension in \Cref{def:extab} has a manifestly-positive combinatorial interpretation like the one given in \Cref{thm:combinatorial-interp}.
% We prove \Cref{thm:refinement-of-ber,,cor:peakenumeration} by proving the statement for every maximal chain~$\cM$ individually.
% \Cref{eq:extending-equality,,eq:peakenumeration}.
% Defining the (\Dfn{combinatorial}) \Dfn{extended $\aaa\bbb$-index} of a \emph{general} finite edge-labeled graded poset of rank~$n$ by
% \[
%   \combextPsi(P;y,\aaa,\bbb) = \sum_{(\cM,E)} y^{\#E}\cdot\StatMon(\cM,E) \ \in \N[y]\langle\aaa,\bbb\rangle\,,
% \]
% means that the proofs of \Cref{thm:refinement-of-ber,,cor:peakenumeration} remain valid. 
% In particular, we obtain
% \[
%   \combextPsi(P;y,\aaa,\bbb)
%   = \omega\big(\combextPsi(P;0,\aaa,\bbb)\big) = \sum_{(\cM,S)} (1+y)^{\# S}\cdot y^{\bout(\cM,S)} \cdot \wt_S \,,
% \]
% where the sum ranges over all maximal chains~$\cM$ and all $\cM$-peak-covering $S \subseteq \{1,\dots,n\}$, where $\wt_S = w_1\dots w_n$ is given in \Cref{eqn:ext-weights}, and where $\bout(\cM,S)$ is the number of positions $i \notin S$ for which $u_i = \bbb$ in $\StatMon(\cM) = u_1\dots u_n$.
% Moreover, $\combextPsi(P;y,\aaa,\bbb)$ is still a polynomial in $\ccc_1 = \aaa+y\bbb, \ccc_2 = \bbb+y\aaa$ and $\ddd = \aaa\bbb+y\bbb\aaa+y\aaa\bbb+y^2\bbb\aaa$, and moreover $2\cdot \combextPsi(P;1,\aaa,\bbb)$ is a direct $\aaa\bbb$-analogue of the \emph{peak enumerator} from~\cite[Definition 7.1]{bergeron-mykytiuk-sottile-vanWilligenburg2000}.
% \end{remark}

\section{Connection to quasisymmetric functions}
\label{sec:connections}

% So far, we have only presented our results for $R$-labeled posets.
% In this section, we consider more general graded posets with arbitrary edge labelings and explore connections to quasisymmetric functions.
% We end with a Schur-positivity conjecture of the map~$\omega$ from \Cref{thm:refinement-of-ber} applied to a Schur function.

% \medskip

\Cref{thm:combinatorial-interp} shows that the extended $\aaa\bbb$-index of an $R$-labeled poset has nonnegative coefficients.
Nonnegativity may fail, however, for posets that do not admit $R$-labelings.
For example, the weak order for the symmetric group $\mathfrak{S}_3$ (the hexagon poset) does \emph{not} admit an $R$-labeling and has extended $\aaa\bbb$-index
\begin{align*}
  \aaa\aaa\aaa &+ (-1+2y)\aaa\aaa\bbb + (1+2y)\aaa\aaa\bbb + y(2 + y^2)\bbb\aaa\aaa + (2y^2 - 1)\aaa\bbb\bbb \\
  &+ (-y^3 + 2y^2) \bbb\aaa\bbb + y^2(2 + y)\bbb\bbb\aaa + y(3y^2 + 2y - 2)\bbb\bbb\bbb\,.
\end{align*}
Using the right-hand side in \Cref{thm:combinatorial-interp}, we define the (\Dfn{combinatorial}) \Dfn{extended $\aaa\bbb$-index} of a finite edge-labeled graded poset~$P$, which is \emph{manifestly positive}, via
\[
  \combextPsi(P;y,\aaa,\bbb) = \sum_{(\cM,E)} y^{\#E}\cdot\StatMon(\cM,E) \ \in \N[y]\langle\aaa,\bbb\rangle\,.
\]
% If $P$ is $R$-labeled, then $\combextPsi$ and $\extPsi$ coincide (thanks to \Cref{thm:combinatorial-interp}), but the advantage of this combinatorial analogue is that $\combextPsi$ is \emph{manifestly positive} for all posets.

While $\combextPsi$ is in general not linked to the Poincaré polynomial, the proofs of \Cref{thm:refinement-of-ber,,cor:peakenumeration} still hold.
In particular, $\combextPsi(P;y,\aaa,\bbb)$ is a polynomial in $\ccc_1 = \aaa+y\bbb, \ccc_2 = \bbb+y\aaa$ and $\ddd = \aaa\bbb+y\bbb\aaa+y\aaa\bbb+y^2\bbb\aaa$.
This means that $2\cdot \combextPsi(P;1,\aaa,\bbb)$ is an $\aaa\bbb$-analogue of the \emph{peak enumerator} from~\cite[Definition 7.1]{bergeron-mykytiuk-sottile-vanWilligenburg2000}.
The remainder of this section is devoted to presenting a conjecture inspired by this specialization.

Let $S = \{s_1 <\dots <s_k\}$ be a subset of $\{1,\dots,n\}$.
The \Dfn{monomial quasisymmetric function}~$M_S$ is the power series
\[
  M_S = \sum_{i_1 < i_2 < \cdots < i_k < i_{k+1}} x_{i_1}^{s_1} x_{i_2}^{s_2 - s_1} \cdots x_{i_k}^{s_k-s_{k-1}}x_{i_{k+1}}^{n+1   - s_k} \quad \in \mathbb{Q}[[x_1,x_2,x_3,\dots]]\,.
\]
Note that~$M_S$ is homogeneous of degree~$n+1$ and---although we surpress it in the notation---implicitly depends on~$n$.
The ring of \Dfn{quasisymmetric functions} $\QSym$ is the (linear) span of $M_{\bullet} = 1$ and all $M_S$ for $n \geq 0$.
% Gessel introduced $\QSym$ to study $P$-partitions and gave a second basis
% \begin{align}
%   \label{eq:FSMT}
%   F_{S} = \sum_{\{1,\dots,n\} \supseteq T \supseteq S} M_T \in \QSym\,,
% \end{align}
% which is related to the monomial quasisymmetric functions via inclusion-exclusion~\cite[Equation~2]{gessel}.
Following~\cite[Section~3]{Ehrenborg-Readdy-transforms}, we define a vector space isomorphism
$\Xi : \QQ\langle\aaa,\bbb\rangle \longrightarrow \QSym$ defined by sending $\wt_T$ to $M_T$.
Using the isomorphism~$\Xi$, we can view the map~$\omega$ from \Cref{thm:refinement-of-ber} as a map from $\QSym$ to $\QSym \otimes \QQ[y]$
given by $F_S \mapsto \omega(F_S) = \Xi\big(\omega(\mon_S)\big)$, where $F_S$ is given in \cite[Equation 2]{gessel}.
In~\cite[Equation~(1.8)]{stembridge-p-partitions}, Stembridge shows how to obtain (skew) Schur functions as $P$-partition enumerators of certain posets given in~\cite[Section~1.3]{stembridge-p-partitions}.
The following conjecture\footnote{This conjecture was exhibited at the \emph{90th Séminaire Lotharingien de Combinatoire} in Bad Boll, Germany in September 2023 in collaboration with Darij Grinberg.} concerning the Schur functions has been verified for all integer partitions of size at most $11$ using \texttt{SageMath}.
\begin{conjecture}
  For any partition $\lambda \vdash n$, the quasisymmetric function $\omega(s_\lambda)$ is symmetric and Schur positive. Specifically, for each $\mu\vdash n$, there exist $c_{\lambda}^{\mu}(y)\in \N[y]$ such that 
  \[
    \omega(s_\lambda) = \sum_{\mu \vdash n} c_{\lambda}^{\mu}(y)\cdot s_{\mu}\,.
  \]
\end{conjecture}


\acknowledgements{
    % This work was initiated at the \emph{Combinatorial Coworkspace} in March 2022 and continued at the conference \emph{Geometry Meets Combinatorics} the following September in Bielefeld.
% We thank the organizers of both events for bringing us together and for fostering collaborative work environments.
We thank
Aram Dermenjian, % for useful discussions regarding support maps for left regular bands,
Richard Ehrenborg,
Darij Grinberg,
Martina Juhnke, and % for clarifying the relationship between $R$-labelings and Cohen-Macaulayness, and
Vic~Reiner for %drawing our attention to the work of Billera-Ehrenborg-Readdy.
useful discussions.
% We believe that they also arrived at the same polynomials in the case of distributive lattices.
}

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