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\DeclareMathOperator{\TN}{\mathrm{TN}}
\newcommand{\zero}{\hat{0}}
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\newcommand{\NN}{\mathbb{N}}
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\title{Chow polynomials of totally nonnegative matrices and posets}

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\author{Petter Brändén\thanks{\href{pbranden@kth.se}{pbranden@kth.se}. PB is a Wallenberg Academy Scholar
  supported by the Knut and Alice Wallenberg
  Foundation, and the G\"oran Gustafsson foundation.}\addressmark{1} \and Lorenzo Vecchi\thanks{\href{lvecchi@kth.se}{lvecchi@kth.se}} \addressmark{1}}

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\address{\addressmark{1}Department of Mathematics, KTH Royal Institute of Technology, Sweden}

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\abstract{Huh-Stevens and Ferroni-Schr\"oter independently conjectured that Hilbert-Poincaré series of Chow rings of  geometric lattices have only real zeros, and Ferroni, Matherne and the second author extended this conjecture to Chow polynomials of Cohen-Macaulay posets. In this paper we address the above conjectures by providing new defining relations and properties of Chow functions of posets and matrices. 
These are used to prove that Chow polynomials of totally nonnegative matrices have only real zeros, which proves the above conjectures for projective and affine geometries, face lattices of cubical polytopes, partition lattices and Dowling lattices, perfect matroid designs, and lattices of flats of paving matroids.  

We also study Chow polynomials of Toeplitz matrices in greater detail, and show how these are related to the combinatorics of binomial and Sheffer posets, as well as to a family of generalized Eulerian polynomials  that have been studied  by e.g. Stanley, Brenti, Stembridge and Shareshian-Wachs.}

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%\resume{\lipsum[2]}

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\keywords{Totally nonnegative matrix, real-rooted polynomial, P\'olya frequency sequence, Chow polynomial, Chow ring, geometric lattice, binomial poset, Sheffer poset}

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\begin{document}
\articlenumber{20}
\maketitle
\section{Introduction}


In their influential work \cite{feichtner-yuzvinsky}, Feichtner and Yuzvinsky introduced the Chow ring of an atomistic lattice $L$, and gave a presentation for it via relations in terms of chains in the lattice. This ring is graded and finite-dimensional, allowing one to study its Hilbert-Poincaré series as a polynomial invariant of the lattice.
When $L$ is a geometric lattice, then the Chow ring satisfies the K\"ahler package \cite{adiprasito-huh-katz}, which implies that its Hilbert series is always palindromic and unimodal. Moreover, for specific classes of geometric lattices, these Hilbert series coincide with well-known families of polynomials; for example  for boolean algebras one recovers the Eulerian polynomials, and for truncations of boolean algebras one recovers the derangement polynomials \cite{hameister-rao-simpson}. For these reasons, the Hilbert series of Chow rings of geometric lattices (or matroids)  have garnered importance on their own and are now known as the \emph{Chow polynomials} of matroids. 

Ferroni-Schr\"oter \cite[Conjecture 8.18]{ferroni-schroter} and Huh-Stevens  \cite[Conjectures 4.1.3 and 4.3.3]{stevens-bachelor} independently  conjectured that Chow polynomials  of matroids have only real zeros. This was recently verified in the uniform case in  \cite[Theorem 1.1]{branden-vecchi}. % and  \cite[Theorem 1.10]{ferroni-matherne-stevens-vecchi}, respectively. 
Ferroni, Matherne and the second author extended the definition of Chow polynomials to arbitrary bounded posets using the framework of Kazhdan-Lusztig-Stanley theory \cite{ferroni-matherne-vecchi}. These are palindromic polynomials that are conjectured to have only real zeros for all Cohen-Macaulay posets and Bruhat intervals (with the appropriate $P$-kernels \cite[Conjectures 4.26 and 6.14]{ferroni-matherne-vecchi}). In this larger context, Hoster and Stump \cite[Theorem 1.1]{hosterstump} proved that Chow polynomials of boolean complexes with nonnegative $h$-vectors are real-rooted.

In this paper we develop a systematic approach to the  study of zeros of Chow polynomials for a class of posets called $\TN$-\emph{posets}, which were introduced in \cite{branden-saud-1,branden-saud-2}. This class includes (rank-selected subposets of) affine and projective geometries, perfect matroid designs, dual partition lattices and dual Dowling lattices. To do so, we define and study Chow polynomials of \emph{totally nonnegative  matrices} ($\TN$-matrices), i.e., matrices whose minors are all nonnegative. Our main result is Theorem \ref{mainman2}, which states that the Chow polynomial of any lower triangular $\TN$-matrix with all diagonal entries equal to one is real-rooted. A direct consequence of this is Theorem~\ref{mainposet}, which says that the Chow polynomial of any (dual) $\TN$-poset is real-rooted. We also use our methods to prove that the Chow polynomial of any paving matroid is real-rooted, Corollary~\ref{cor:main paving}. 

We mention that Coron, Ferroni and Li \cite{coron-ferroni-li} independently proved that the Chow polynomials of  so called UMEL-shellable posets are real-rooted. While there seems to be great intersection between this class and the one of TN posets, a clear relation between the two is at the moment only conjectured, see \cite[Conjecture~4.3]{branden-saud-2} and \cite[Section~8]{coron-ferroni-li}.

For proofs of the results presented in this extended abstract, we refer to \cite{branden-vecchi-2}. \\


\section{The incidence algebra and Chow functions}\label{section: Chow posets}
In this section we recall the construction of Chow functions and Chow polynomials of partially ordered sets (posets). For undefined poset terminology, we refer to \cite{stanley-ec1}. Recall that an interval of a poset is a subposet of $P$ of the form $[x,y]= \{z \in P : x\leq z \leq y\}$, where $x,y\in P$. 
All posets $P$ considered in this paper are \emph{locally finite}, i.e., each interval of $P$ is finite. We also say that a poset is \emph{bounded} if it has unique least and largest elements, which we denote by $\zero$ and $\one$, respectively.
Given a poset $P$, a \emph{weak rank function} is a function $\rho: P\times P \to \NN$ such that 
\begin{itemize}
    \item $\rho(x,y) = \rho_{x,y} > 0$ if and only if $x < y$, and
    \item  $\rho_{x,y} = \rho_{x,z} + \rho_{z,y}$, for all  $x\leq z \leq y$ in $P$.
\end{itemize}
A \emph{weakly ranked} poset consists of a pair $(P,\rho)$, where $\rho$ is a weak rank function for $P$. By slight abuse of notation, we say that $P$ is a weakly ranked poset, when this does not create confusion.
If the poset has a least element $\zero$, we write $\rho(x):= \rho_{\zero,x}$ for $x \in P$ and call this the \emph{(weak) rank} of $x$ in $P$. If $\rho_{x,y} =1$ whenever $y$ covers $x$, then we say that $P$ is \emph{graded} (or \emph{ranked}). Moreover if $P$ is bounded, then we say that $P$ has (weak) rank $\rho(\one)$. 

The \emph{incidence algebra}, $I(P)$,  of a poset $P$ is the free $\RR[t]$-module spanned by the intervals of $P$.
More explicitly, an element $f$ associates to every interval $[x,y]$ of $P$ an element in $\RR[t]$ which we denote by $f_{x,y}$. The product (\emph{convolution}) is defined by
\[
(fg)_{x,y} = \sum_{x\leq z \leq y}f_{x,z}\cdot g_{z,y}.
\]
For a polynomial $f \in \RR[t]$ of degree at most $n$, we define 
$
\I_n(f)(t) = t^nf(t^{-1})
$
and say that $f$ is \emph{palindromic with center of symmetry} $n/2$, if ${\I_n(f) = f}$. 
Let $I_\rho(P)$ be the subalgebra of $I(P)$ consisting of functions $f$ such that $\deg f_{x,y} \leq \rho_{x,y}$ for all $x,y \in P$, and  define ${\I:I_\rho(P) \to I_\rho(P)}$ by 
$
\mathcal I (f)_{x,y} = \mathcal I_{\rho_{x,y}}(f_{x,y})$. 
Hence, $\I$ is an involution and an automorphism, i.e., $\I^2=\delta$ and  $\I(fg)=\I(f)\I(g)$ for all $f,g \in I_\rho(P)$. 

Given a weakly ranked poset $P$, an element $\kappa$ in  $I_\rho(P)$ is called a \emph{$P$-kernel} if
$\kappa_{x,x} = 1$ for each $x \in P$, and 
$\kappa^{-1} = \mathcal I(\kappa)$. Equivalently, $\kappa \in I_\rho(P)$ is a $P$-kernel if and only if $\kappa = g^{-1}\mathcal I (g)$ for some $g\in I(P)$ such that
    \begin{equation}\label{condition for kls}
    g_{x,x}=1 \mbox{ for each } x \in P, \ \ \mbox{ and }  \ \ \deg g_{x,y}< {\rho_{x,y}}/{2} \text{ for every $x < y$ in $P$}, 
    \end{equation}
see \cite[Theorem~6.2]{brenti-kls}, \cite[Proposition 2.5]{proudfoot-kls}. 
The function $g$ is called the \emph{left Kazhdan-Lusztig-Stanley function} associated to $\kappa$. 
Following \cite[Definition 3.2]{ferroni-matherne-vecchi}, we define the \emph{reduced $P$-kernel} to be the function $\overline{\kappa} = (\kappa-t\delta)/(t-1)$ in $I_\rho(P)$ and the \emph{Chow function} associated to $\kappa$ (or $g$) as the element $(-\overline{\kappa})^{-1}$ in $I_\rho(P)$.  

For every $n \in \NN$, define a family of operators $\SSS_n$ on polynomials of degree at most $n$ in $\RR[t]$ by 
$
\SSS_n(f )= (\I_n(f) - f)/(t-1).
$
The following theorem provides an alternative definition of the Chow function.
\begin{theorem}\label{thm: def H and d}
    Let $g \in I(P)$ be an element that satisfies \eqref{condition for kls}. Then there exist unique functions $H$ and $d$ in $I_\rho(P)$ for which
\begin{multicols}{2}
    \begin{itemize}
        \item[(i)] $d_{x,x} = 1$ for each  $x \in P$,
        \item[(ii)] $\I(H) = tH + (1-t)\delta$, 
        \item[(iii)]  $\I(d) = d$,   
        \item[(iv)]  $H = dg$.
    \end{itemize}
    \end{multicols}
    Moreover, $H$ is the Chow function associated to $g$, and the polynomials $d_{x,y}$ satisfy the recursion 
    \begin{equation}\label{d-recu}
d_{x,y} = t\SSS_{\rho_{x,y}-1}\left(\sum_{x \leq z <y}d_{x,z}g_{z,y}\right),  \quad x<y. 
\end{equation}
\end{theorem}

We call the unique function $d$ achieved by Theorem \ref{thm: def H and d} the \emph{Chow-derangement function} associated to $g$. If the poset is bounded, we call $H_P = H_{\zero,\one}$ the \emph{Chow polynomial} of $P$ (with respect to $g$) and $d_P= d_{\zero,\one}$ the \emph{Chow-derangement polynomial} of $P$.


In this paper we will be mostly concerned with a special class of functions $g$, where $g_{x,y} \in \RR$ for all $x\leq y$ and $g_{x,x} = 1$ for each $x \in P$. We call such functions \emph{scalar}. An example of a scalar function is $\zeta$, the \emph{zeta function} of $P$, which is defined by $\zeta_{x,y} =1$ for all $x \leq y$ in $P$. 

\begin{example}
    When $g = \zeta$, the corresponding $P$-kernel is the \emph{characteristic function} $\chi$, which associates to each interval its characteristic polynomial. Hence, the associated Chow function $H$ was given the name of \emph{characteristic Chow function} in \cite{ferroni-matherne-vecchi}. When $P$ is a geometric lattice, the polynomial $H_P$ coincides with the Hilbert series of the Chow ring of $P$ \cite[Theorem 1.4]{ferroni-matherne-stevens-vecchi}. The polynomials $H_P$ are $\gamma$-positive for Cohen-Macaulay posets \cite[Theorem~1.4]{ferroni-matherne-vecchi}, and are conjectured to be real-rooted for every Cohen-Macaulay poset \cite[Conjecture~1.5]{ferroni-matherne-vecchi}.
\end{example}

\begin{example}\label{H-Eul-ex}
    When $P$ is a boolean algebra of rank $r$ and $g = \zeta$, then all intervals of the same rank in $P$ are isomorphic, from which it follows that $H_{x,y}= H_{\rho(y)-\rho(x)}$ and $d_{x,y}= d_{\rho(y)-\rho(x)}$ for some polynomials $\{H_n\}_{n=0}^r$ and $\{d_n\}_{n=0}^r$. Theorem \ref{thm: def H and d} then says that these two families of polynomials are the unique ones that satisfy $H_0=d_0=1$ and, for $1\leq n \leq r$, $H_n = \sum_{k=0}^n\binom{n}{k}d_k$, $\I_n(d_n)= d_n$ and $\I_{n}(H_n)=tH_n$. 
    These properties are known to hold for the \emph{derangement polynomials} $d_n = \sum_{\sigma \in \mathfrak{D}_n }t^{ \mathrm{exc}(\sigma) }$,
    which count \emph{derangements} (fixed point free permutations) in the symmetric group $\mathfrak{S}_n$ by the number of \emph{excedances} $\mathrm{exc}(\sigma)=|\{i : \sigma(i)>i\}|$, and the \emph{Eulerian polynomials} $A_n = \sum_{\sigma \in \mathfrak{S}_n }t^{ \mathrm{exc}(\sigma) }$, see \cite[Corollary 1]{Brenti-90}. Hence we deduce that $d_{x,y}$ and $H_{x,y}$ are the traditional derangement polynomials and Eulerian polynomials, respectively. 
\end{example}

\begin{remark}
    Given a geometric lattice, one may also define its \emph{augmented} Chow ring. In analogy with that, one defines an \emph{augmented Chow polynomial} $G_P$ for any poset $P$ so that, when $P$ is a geometric lattice, $G_P$ computes the Hilbert series of the corresponding augmented Chow ring. By \cite[Corollary 4.6]{ferroni-matherne-vecchi} $G_P$ is itself a Chow polynomial associated to the poset $\operatorname{aug}(P)$, which is built by adding a new least element to $P$. Our results then naturally extend to the augmented case, see \cite{branden-vecchi-2}. 
\end{remark}

\section{Chow polynomial of totally nonnegative matrices}\label{section chow matrix}
Let $R=(r_{n,k})_{n,k=0}^N$, where $N \in \NN \cup \{\infty\}$, be a lower triangular matrix with all diagonal entries equal to one. Consider the chain $[0,N]=\{0<1<\cdots<N\}$ graded by the rank function $\rho(n) = n$. Then the matrix $R$ corresponds to an element $g = g[R]$ in the incidence algebra of $[0,N]$ defined by
\[
g_{k,n} = r_{n,k}, \quad \mbox{ for all } 0\leq k \leq n \leq N.
\]
Clearly $g$ satisfies \eqref{condition for kls} and therefore we can define the corresponding Chow and Chow-derangement polynomials. The following corollary provides the defining relations for these polynomials. 

\begin{corollary}\label{cor: H and d for matrices}
 Let $R=(r_{n,k})_{n,k=0}^N$, $N \in \NN \cup \{\infty\}$, be a lower triangular matrix with all diagonal entries equal to one. There are unique polynomials $\{d_n\}_{n=0}^N$ and $\{H_n\}_{n=0}^N$ for which 
 \begin{multicols}{2}
    \begin{itemize}
    \item[(i)] $d_0=1$, 
    \item[(ii)]$\I_n(H_n)=tH_n$, 
    \item[(iii)] $\I_n(d_n) =d_n$,  
    \item[(iv)] $H_n=\sum_{k=0}^n r_{n,k} d_k$,  
\end{itemize} 
 \end{multicols}
\noindent for each $1 \leq n \leq N$.

Moreover $d_n=d_{0,n}[R]$ and $H_n=H_{0,n}[R]$, $n \leq N$,  are the Chow-derangement polynomials and the Chow polynomials associated to $g[R]$, respectively, and 
\begin{equation}\label{dn-recu}
d_n= t\SSS_{n-1}\left(\sum_{k=0}^{n-1} r_{n,k} d_k \right), \quad 1 \leq n \leq N. 
\end{equation}
\end{corollary}

%In Section \ref{section toeplitz} we will need the following result which tells us how the 
%Chow polynomials behave under conjugation by a diagonal matrix. 


%\section{Totally nonnegative matrices and their Chow-polynomials}\label{TNChowsection}

The main result of this section is Theorem \ref{mainman2}, which says that the Chow polynomial of any lower triangular and totally nonnegative matrix with all diagonal entries equal to one is real-rooted. 
%\subsection{Totally nonnegative and resolvable matrices}
We start by collecting some notation and results from \cite{branden-saud-1}. Recall that a matrix with real entries is \emph{totally nonnegative} (or $\TN$) if all of its minors are nonnegative. 
\begin{definition}
\label{resolv}
Let $R=(r_{n,k})_{n,k=0}^N$, where $N \in \NN \cup\{\infty\}$,  be a lower triangular matrix with real entries whose diagonal entries are all equal to one, and let $R_n = \sum_{k=0}^n r_{n,k} t^k$ be the generating polynomial of the $n$th row. 
The matrix $R$ is called \emph{resolvable} if there is an array  $(\lambda_{n,k})_{0\leq k\leq n <N}$ of nonnegative numbers, and an array of monic polynomials $(R_{n,k})_{0\leq k \leq n \leq N}$ in $\RR[t]$ for which 
\begin{itemize}
\item $R_{n,0}=R_n$ and $R_{n,n} =t^n$ for each $0\leq n \leq N$,
\item $t^k$ divides  $R_{n,k}$ for all $0 \leq k \leq n \leq N$, and 
\item if $0\leq k \leq n <N$, then 
%\begin{equation}\label{r-pasc}
$R_{n+1,k}=R_{n+1,k+1}+ \lambda_{n,k} R_{n,k}$. 
%\end{equation}
\end{itemize}
If the matrix $R$ is resolvable, then we say that the polynomials $(R_{n,k})_{0\leq k \leq n \leq N}$ \emph{resolve} $R$.
%If all $\lambda_{n,k}$ are positive, then we say that $R$ is \emph{strictly resolvable}. 
\end{definition}

\begin{example}
The identity matrix is resolvable, with $R_{n,k}=t^n$ and $\lambda_{n,k}=0$ for all $0 \leq k \leq n$. 
Pascal's triangle $(\binom n k)_{n,k=0}^\infty$ is resolvable, with $R_{n,k}=t^k(1+t)^{n-k}$ and $\lambda_{n,k}=1$ for all $0 \leq k \leq n$. 
For more examples we refer to \cite{branden-saud-1,branden-saud-2}. 
\end{example}
The next theorem, proved in \cite[Theorem 2.6]{branden-saud-1}, characterizes resolvability in terms of totally nonnegative matrices. 
\begin{theorem}\label{eqcon}
    Let $R$ be a lower triangular matrix with all diagonal entries equal to one. Then  $R$ is resolvable if and only if $R$ is $\mathrm{TN}$. 
\end{theorem}


\subsection{\texorpdfstring{Interlacing zeros and $\mathcal I_n$-interlacing sequences of polynomials}{Interlacing zeros and In-interlacing sequences of polynomials}}
Recall that a polynomial $p \in \RR[t]$ is called \emph{real-rooted} if all of its zeros are real. %For technical reasons, the zero polynomial is considered to be real-rooted. 
Suppose $p$ and $q$ are two real-rooted polynomials in $\RR_{\geq 0}[t]$ with zeros $\cdots \leq \beta_2 \leq \beta_1$ and $\cdots \leq \alpha_2 \leq \alpha_1$, respectively.
The zeros \emph{interlace} if 
$$
  \cdots \leq \beta_2 \leq \alpha_2 \leq \beta_1\leq \alpha_1 . 
$$
We then write $p \prec q$. 
For technical reasons we consider the identically zero polynomial to be real-rooted and write $0 \prec p$ and $p \prec 0$ for any other real-rooted polynomial $p$. 
A sequence of polynomials $f_1, \ldots, f_m$ in $\RR_{\geq 0}[t]$ is called \emph{interlacing} if $f_i \prec f_j$ for all $i<j$. 


%We will now define a central notion of this paper. 
The following notion will be fundamental in the proofs of our main results.
\begin{definition}\label{def: I_n-int}
    We say that a sequence $f_1, f_2, \ldots, f_m $ of polynomials in $\RR_{\geq 0}[t]$  of degree at most $n$ is  $\I_n$-\emph{interlacing} if 
$$
f_1, f_2, \ldots, f_m, \I_n(f_m), \I_n(f_{m-1}), \ldots, \I_n(f_1)
$$
is an interlacing sequence of polynomials. 
\end{definition}

\subsection{Chow-deranged maps}\label{maps}
Let  $R=(r_{n,k} )_{n,k=0}^N$, where $N \in \NN\cup\{\infty\}$, be a lower triangular matrix, with entries in $\RR$, and  with all diagonal entries equal to one. Let further $\RR_N[t]$ be the linear space of all polynomials in $\RR[t]$ of degree at most $N$. We define the $\RR$-linear map 

$$
\DD =\DD_R : \RR_N[t] \to \RR_N[t], \qquad \DD(t^n)= d_n
$$
where $d_n$ is the Chow-derangement polynomial associated to $R$. This map is called the \emph{Chow-deranged map}. Notice that (iv) in Corollary \ref{cor: H and d for matrices} translates to
$
H_n = \DD(R_n)$. 
\begin{remark}
    The deranged map was first studied by the first author and Solus \cite{branden-solus}, who proved Theorem~\ref{mainman} below for the case when $R=(\binom n k )_{n,k=0}^\infty$. %The Chow-Eulerian map for the case when $R=R_0$ was first considered by Brenti \cite{Brenti89}, who conjectured that $\A_{R_0}(f)$ is real-rooted whenever all zeros of $f \in \RR[t]$ are real and nonpositive. This conjectured was disproved in \cite{branden-jochemko}, where it was conjectured that $\A_{R_0}(f)$ is real-rooted whenever $f$ has a nonnegative expansion in $\{t^k(1+t)^{n-k}\}_{k=0}^n$. This conjecture was proved by Athanasiadis in \cite{athanasiadis-eulerian2}. 
\end{remark}

%Theorem~\ref{mainmanA} below generalizes Athanasiadis result to any resolvable matrix.
Suppose $R$ is resolvable, and define 
$
d_{n,k} = \DD(R_{n,k})$,  $0\leq k \leq n \leq N$. 


\begin{theorem}\label{mainman} Let $R$ be a resolvable matrix, and $0\leq n \leq N$. Then 
$$
d_{n,0}, d_{n,1}, \ldots, d_{n,n}
$$
is an $\I_n$-interlacing sequence. 
Moreover if $f= \sum_{k=0}^n h_k R_{n,k}$, where $h_k \geq 0$ for each $k$, then $\DD(f)$ is real-rooted and $d_{n,0} \prec \DD(f) \prec d_{n,n}$. 
\end{theorem}


The first important consequence of Theorem \ref{mainman} establishes the real-rootedness of the Chow polynomials and Chow-derangement polynomials of any lower triangular $\TN$-matrix with all diagonal entries equal to one. 
\begin{theorem}\label{mainman2}
Let $R = (r_{n,k})_{n,k=0}^N$ be a lower triangular $\TN$-matrix with all diagonal entries equal to one. Then the Chow polynomials $\{H_n\}_{n=0}^N$ and the Chow-derangement polynomials $\{d_n\}_{n=0}^N$ are real rooted. 

Moreover $H_n \prec d_n$, $H_n \prec H_{n+1}$ and  $d_n \prec d_{n+1}$ for each $n$. 
\end{theorem}

\section{\texorpdfstring{Chow polynomials of $\mathrm{TN}$-posets}{Chow polynomials of TN-posets}}\label{section real zeros TN posets}
%\subsection{Weak-rank uniform posets}
We shall now see that the Chow polynomials of certain matrices are actually characteristic Chow polynomials of posets, by following the construction in \cite[Section 5]{branden-saud-1}. We say that a weakly ranked poset $P$ with a least element $\zero$ is \emph{weak-rank uniform} if for any $x$ and $y$ in $P$ with $\rho(x) = \rho(y)$, 
\begin{equation}\label{wru}
|\{z \leq x : \rho(z) = k\}| = |\{z \leq y : \rho(z) = k\}|, \quad \text{ for each }0\leq k \leq \rho(x).
\end{equation}
If $\rho(x) = n$, we define $r_{n,k}=r_{n,k}(P) = |\{z \leq x : \rho(z) = k \}|$, and write $R = R(P) = (r_{n,k})_{n,k\geq 0}$. Notice that $r_{n,0} = r_{n,n} = 1$ for each $n$. If $P$ is graded and satisfies \eqref{wru}, then we say that $P$ is \emph{rank-uniform}. 
\begin{proposition}\label{matrix-poset}
    Let $P$ be a weak-rank uniform poset, and let $R = (r_{n,k}(P))_{n,k=0}^N$ be the corresponding matrix, where $N\in \NN \cup \{\infty\}$. If $n \leq N$, then $H_n[R]$ is equal to the characteristic Chow polynomial $H_{[\zero,x]}$, where $x$ is any element of rank $n$.
\end{proposition}

In this paper, a weakly rank-uniform poset $P$ is called a $\TN$-\emph{poset} if the matrix $R(P)$ is $\TN$. Below is a list of examples of $\TN$-posets, for proofs see \cite{branden-saud-1, branden-saud-2}. 
\begin{itemize}[leftmargin=6mm]
\item[\textbf{a}.] Boolean cell complex with nonnegative $h$-vectors. For example  Cohen-Macaulay simplicial complexes and face lattices of simplicial polytopes. 
\item[\textbf{b}.] Cubical complexes with nonnegative cubical $h$-vectors \cite{adin1996new}. These include face lattices of cubical polytopes, i.e., polytopes for which all faces are hypercubes. 
\item[\textbf{c}.] $q$-poset with nonnegative $h$-vectors \cite{alder2010q}. For example shellable $q$-complexes \cite{Ghorpade}. 
\item[\textbf{d}.] Perfect matroid designs \cite{Deza}, i.e., rank uniform geometric lattices. These include (truncations of) projective geometries and affine geometries. 
\item[\textbf{e}.] Dual of Dowling lattices \cite{dowlingGroups}. In particular, the dual of partition lattices. 
    \end{itemize}
Recall that if $P$ is a weakly ranked poset of rank $r$ and $S \subseteq [r-1]$, then 
$$
P_S= \left\{ x \in P : \rho(x) \in S \cup \{0,r\}\right\}
$$
is the \emph{rank selected subposet} induced by $S$. Since submatrices of $\TN$-matrices are $\TN$ it follows that the class of $\TN$-posets is closed under rank selection. Hence any rank-selected poset of any poset listed above is $\TN$. 
\begin{theorem}\label{mainposet}
Suppose $P$ is a $\TN$-poset, or the dual of a $\TN$-poset, of rank $n$. Then the characteristic Chow polynomial $H_P$ is real-rooted. 
\end{theorem}


If we apply Theorem \ref{mainposet} to \textbf{a} above, then we recover \cite[Theorem 1.1]{hosterstump}. 
Theorem \ref{mainposet} for the case \textbf{e} above implies in particular that Chow polynomials of Dowling lattices are real-rooted.%\footnote{Coron, Ferroni, and Li informed us that they independently found a different proof for the real-rootedness of Chow polynomials of Dowling lattices.}.



\section{Chow polynomials of lattices of flats of paving matroids}\label{pavingsection}
Let $P$ be a graded, rank uniform and bounded poset of rank $n$ and let $0\leq d <n$. Consider a subset $\mathcal H \subseteq P$ of pairwise incomparable elements such that $d < \rho(x) < n$ for each $x \in \mathcal H$. Let $P(d,\mathcal H)$ be the weakly ranked poset of rank $n$ obtained by adjoining $\one$ and $\mathcal H$ to the set $\{x \in P \mid \rho(x) \leq d\}$ and by declaring all elements in $\mathcal H$ to be of rank $d+1$ and $\rho(\one) = d+2$. Notice also that if 
\begin{equation}\label{paving graded condition}
    \text{for each $x$ of rank $\rho(x) \leq d$ there exists a $y \in \mathcal H$ such that $x\leq y$}, 
\end{equation}
then $P(d,\mathcal H)$ is graded again.
If $P$ is a boolean algebra on $n$ elements and $\mathcal H$ satisfies  \eqref{paving graded condition}, then $P(d,\mathcal H)$ is a paving geometric lattice of rank $d+2$.
\begin{theorem}\label{thm:main paving}
    Let $P$ be a graded $\TN$-poset of rank $n$ and $P' = P(d,\mathcal H)$ be as above. Then the  polynomials $d_{P'}$ and  $H_{P'}$ are real-rooted.
\end{theorem}


\begin{corollary}\label{cor:main paving}
    If $L$ is the lattice of flats of a  paving matroid, then the polynomials $d_{L}$ and $H_{L}$ are real-rooted.
\end{corollary}


    

\section{Toeplitz matrices}\label{section toeplitz}
In this section we will focus on the special case when $R$ is a Toeplitz matrix. 
Let $\{a_n\}_{n=0}^\infty$ be a sequence of real numbers, where $a_0=1$.  Associate to $\{a_n\}_{n=0}^\infty$  the lower triangular \emph{Toeplitz matrix}
$R=(a_{n-k})_{n,k=0}^\infty$, 
where $a_m=0$ if $m<0$. 

Consider the formal power series 
$f= \sum_{n=0}^\infty a_nz^n$ in $\RR[[z]]$, and define formal power series in $\RR[t][[z]]$ by 
\begin{alignat*}{2}
 D(z,t) &= \sum_{n =0}^\infty d_n(t) z^n, \quad 
 H(z,t) &= \sum_{n=0}^\infty H_n(t) z^n, \\
 %A(z,t) &= \sum_{n =0}^\infty A_n(t) z^n, \quad 
  %G(z,t) &= \sum_{n=0}^\infty G_n(t) z^n. 
  \end{alignat*}
where $d_n(t), H_n(t),  %A_n(t)$ and $G_n(t)
$  are the Chow-derangement polynomials and Chow polynomials %, %Chow-Eulerian polynomials  and augmented Chow polynomials
associated to $R$, respectively. 

\begin{theorem}\label{fourform}
Let $f= \sum_{n=0}^\infty a_nz^n$ be a formal power series in $\RR[[z]]$, where $a_0=1$. Then 
\begin{align*}
D(z,t) &= \sum_{n =0}^\infty d_n(t) z^n= \frac {1-t}{f(tz)-tf(z)}=  \frac  1 {1- \sum_{n \geq 2 } a_n(t+t^2+\cdots +t^{n-1})z^n},  \\
H(z,t) &= \sum_{n=0}^\infty H_n(t) z^n= \frac {(1-t)f(z)}{f(tz)-tf(z)}. \\
%A(z,t) &= \sum_{n =0}^\infty A_n(t) z^n= \frac {(1-t)f(tz)}{f(tz)-tf(z)}= 1-t+tH(z,t),  \\
%G(z,t) &= \sum_{n=0}^\infty G_n(t) z^n.
\end{align*}
\end{theorem}


A sequence $\{a_n\}_{n=0}^\infty$ of real numbers is a \emph{P\'olya frequency sequence} if the Toeplitz matrix $R=(a_{n-k})_{n,k=0}^\infty$ is $\TN$. 
P\'olya frequency sequences were characterized by Aissen, Schoenberg,  Whitney and Edrei \cite{ASW} as follows. 

\begin{theorem}\label{ASWE}
    A sequence $\{a_n\}_{n=0}^\infty$ of real numbers is a P\'olya frequency sequence if and only if its generating function is of the form 
    \begin{equation}
    \label{PFS}
     f=   \sum_{n=0}^\infty a_n z^n = C z^N e^{\gamma z} \prod_{i=1}^\infty \frac {1+ \alpha_iz} {1- \beta_iz},
    \end{equation}
    where $C, \gamma, \alpha_i, \beta_i$ are nonnegative real numbers, $N \in \NN$, and $\sum_{i=1}^\infty (\alpha_i+\beta_i)<\infty$. 
\end{theorem} 

Applying Theorem \ref{mainman2} to Toeplitz matrices produces families of real-rooted polynomials associated to any series $f$ of the form \eqref{PFS}. In particular if $f(z)=e^z$ this implies the real-rootedness of Eulerian and derangement polynomials. 

\begin{theorem}\label{PFSR}
Suppose $f$ is a power series as in \eqref{PFS}. Then the polynomials $d_n(t), H_n(t)%,  A_n(t)$ and $G_n(t)
$, $n\geq 0$,  defined via $f$ by the identities in Theorem~\ref{fourform} 
are all real-rooted. 
\end{theorem}


\subsection{Binomial and Sheffer posets}\label{binopo}
 Recall \cite{DRS, stanley1976binomial,stanley-ec1} that a  \emph{binomial poset} is a  locally finite poset $P$ for which  there exists an infinite chain in $P$, each interval of $P$ is graded, and 
 there exists a function $B : \NN \to \NN$, called the \emph{factorial function} of $P$, such that the number of maximal chains in any interval $[x,y]$ in $P$ is equal to $B(\rho(x,y))$. 

Hence a binomial poset $P$ is rank uniform with 
\begin{equation}\label{rank-B}
    r_{n,k}(P) = \frac {B(n)} {B(k)\cdot B(n-k)},  \quad  0 \leq k \leq n, 
\end{equation}
since each maximal chain in $[\zero, x]$, $\rho(x)=n$,  passes through a unique element of rank $k$. 

\begin{theorem}\label{th: bin-pos-ser}
Let $P$ be a binomial poset with factorial function $B$, and let 
$$
b(z) = \sum_{n=0}^\infty \frac {z^n}{B(n)}.
$$
The generating functions for the various Chow-polynomials of $P$ have the following expressions.
\begin{alignat*}{2}
\sum_{n =0}^\infty d_n(t) \frac {z^n}{B(n)} &= \frac {1-t}{b(tz)-tb(z)}, \quad \quad 
\sum_{n=0}^\infty H_n(t) \frac {z^n}{B(n)} &= \frac {(1-t)b(z)}{b(tz)-tb(z)}.
%\sum_{n =0}^\infty A_n(t) \frac {z^n}{B(n)} &= \frac {(1-t)b(tz)}{b(tz)-tb(z)}, \quad  \quad 
%\sum_{n=0}^\infty G_n(t) \frac {z^n}{B(n)} &= \frac {(1-t)b(tz)\cdot b(z)}{b(tz)-tb(z)}.
\end{alignat*}
\end{theorem}   

\begin{example}
Let $\mathbb{F}_q$ be a field with $q$ elements, and let $V(q)$ be the free $\mathbb{F}_q$-linear space over the set $\{e_1, e_2, \ldots\}$.  The \emph{projective geometry}
$\mathbb{B}(q)$ is the lattice of all finite dimensional subspaces of $V(q)$.  Then $\BB(q)$ is binomial with factorial function given by 
$$\mathbf{(n)!}=1 \cdot(1+q) \cdots (1+q+\cdots + q^{n-1}).$$    
 By Theorem~\ref{th: bin-pos-ser}, 
$$
\sum_{n=0}^\infty  H_n(t) \frac {z^n}{\mathbf{(n)!}}  = \frac {(1-t)e_q(z)}{e_q(tz)-te_q(z)}, 
 $$
 where $e_q(z)= \sum_{n=0}^\infty z^n/\mathbf{(n)!}$ is the $q$-\emph{exponential function}.
 
 For $\sigma \in \mathfrak{S}_n$, let ${\textrm{maj}(\sigma)= \sum_{i\in \mathrm{D}(\sigma)} i}$. From the work of Shareshian and Wachs \cite{Shareshian-Wachs-10} it follows that 
 $$
 H_n(t) = \sum_{\sigma \in \mathfrak{S}_n} q^{\textrm{maj}(\sigma)-\textrm{exc}(\sigma)} t^{\textrm{exc}(\sigma)}, 
 $$
 the $q$-analog, $A_n(q,t)$, of the Eulerian polynomial studied %by Shareshian and Wachs 
 in \cite{Shareshian-Wachs-10}. 
 This was first proved by Hameister, Rao and Simpson \cite{hameister-rao-simpson}. 
\end{example}
 %Moreover, it follows similarly from Theorem~\ref{th: bin-pos-ser} that the augmented Chow polynomials, $G_n(t)$, of $\BB(q)$ are the $q$-binomial Eulerian polynomials $\widetilde{A}_n(q,t)$ studied by Shareshian and Wachs in \cite{shareshian-wachs-2018}. 

%Since $\BB(q)$ is a $\TN$-poset we know that the polynomials $A_n(q,t)$ and  $\widetilde{A}_n(q,t)$,  $n \in \NN$, are all real-rooted.

Ehrenborg and Readdy generalized the notion of binomial posets in \cite{Ehr-Rea}. A poset $P$ is called a \emph{Sheffer poset} if there are two functions $B : \NN \to \NN$ and $C: \NN \to \NN$ such that 
$P$ is locally finite with a least element $\zero$, $P$ contains an infinite chain,  
each interval in $P$ is graded, %and hence $P$ has a rank function $\rho : P \to \NN$,  
the number of maximal chains in $[\zero, x]$ is equal to $C(\rho(x))$, for each $x \in P$, and  
if $\zero < x \leq y$, then the number of maximal chains in $[x,y]$ is equal to $B(\rho(x,y))$. 

The functions $B$ and $C$ are called the \emph{factorial functions} associated to $P$. Hence Sheffer posets are rank uniform, and binomial posets are the Sheffer posets for which $B=C$. 

\begin{theorem}\label{th: shef-pos-ser}
Let $P$ be a Sheffer poset with factorial functions $B$ and $C$, and let 
$$
b(z) = \sum_{n=0}^\infty \frac {z^n}{B(n)} \quad \mbox{ and } \quad c(z) = \sum_{n=0}^\infty \frac {z^n}{C(n)}. 
$$
Then 
\begin{align*}
\sum_{n =0}^\infty d_n(t) \frac {z^n}{C(n)} &= 1+\frac {1-t}{b(tz)-tb(z)}-   \frac {c(tz)-tc(z)}{b(tz)-tb(z)}, \\
\sum_{n=0}^\infty H_n(t) \frac {z^n}{C(n)} &= \frac {(1-t)b(z)}{b(tz)-tb(z)}+ \frac {c(z)\cdot b(tz)-c(tz)\cdot b(z)}{b(tz)-tb(z)}.\\
%\sum_{n =0}^\infty A_n(t) \frac {z^n}{C(n)} &= 1+ t \frac {c(z)-c(tz)}{b(tz)-tb(z)}, \\
%\sum_{n=0}^\infty G_n(t) \frac {z^n}{C(n)} &=  \frac {c(z)\cdot b(tz)-tc(tz)\cdot b(z)}{b(tz)-tb(z)}.
\end{align*}
\end{theorem}


\begin{example}\label{Ex: Sheffer} 
The \emph{affine geometry} $\mathbb{A}(q)$, ordered by inclusion, may be defined as 
$
\mathbb{A}(q)=  \{ U \setminus H : U \in \mathbb{B}(q)\}, 
$
where $H$ is the subspace of $V(q)$ spanned by $\{e_2,e_3,\ldots\}$. The poset $\mathbb{A}(q)$ is Sheffer (see \cite[Proposition~6.2.5]{Oxley}).  The factorial functions of $\mathbb{A}(q)$ are 
$
B(n)= \mathbf{(n)}!$ and $C(n) = q^{n-1} \cdot \mathbf{(n-1)}!$ . 
Then 
$$
\sum_{n\geq 0} H_{n+1}(t) \frac {z^n}{q^n \mathbf{(n)}!}= \frac {e_q(z/q)\cdot e_q(tz)-te_q(tz/q)\cdot e_q(z)}{e_q(tz)-te_q(z)}. 
$$
\end{example}
%ds3527


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