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% This is the version that Yuval and Vic started on July 11, 2017
%
%\documentclass[11pt]{amsart}
%\usepackage[margin=1in,letterpaper,portrait]{geometry}
\usepackage{ytableau}
\ytableausetup{notabloids}
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\marginpar{\textsc{ToDo}} \framebox{\begin{minipage}[c]{0.95
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 #1 \end{minipage}}\vspace{5 mm}\par}

\newcommand\comp[2]{\alpha{(#2,#1)}}
\newcommand\ccomp[2]{\alpha^\cyc{(#2,#1)}}


\newcommand{\CC}{{\mathbb {C}}}
\newcommand{\QQ}{{\mathbb {Q}}}
\newcommand{\RR}{{\mathbb {R}}}
\newcommand{\ZZ}{{\mathbb {Z}}}

\newcommand{\Cyl}{\operatorname{Cyl}}
\newcommand{\Tor}{\operatorname{Tor}}

\newcommand{\then}{\Rightarrow}

\newcommand{\Aut}{{\operatorname{Aut}}}
\newcommand{\Des}{{\operatorname{Des}}}
\newcommand{\cDes}{{\operatorname{cDes}}}
\newcommand{\cyc}{{\operatorname{cyc}}}
\newcommand{\cdes}{{\operatorname{cdes}}}
\newcommand{\des}{{\operatorname{des}}}
\newcommand{\ch}{{\operatorname{ch}}}
\newcommand{\shape}{{\operatorname{shape}}}
\newcommand{\SYT}{{\operatorname{SYT}}}
\newcommand{\co}{{\operatorname{co_n}}}
\newcommand{\cc}{{\operatorname{cc}_n}}

\newcommand{\Comp}{{\operatorname{Comp}}}
\newcommand{\cComp}{{\operatorname{cComp}}}
\newcommand{\ccon}{{\operatorname{cyc-comp}(n)}}

\newcommand{\one}{{\mathbf{1}}}

\newcommand{\cs}{{\tilde{s}}}

\newcommand{\xx}{{\mathbf{x}}}
\newcommand{\ttt}{{\mathbf{t}}}
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\newcommand{\MMM}{{\mathcal{M}}}
\newcommand{\OOO}{{\mathcal{O}}}
\newcommand{\TTT}{{\mathcal{T}}}

\newcommand{\SteinbergTorus}{{\widetilde{\Delta}}}

\newcommand{\wcDes}{{\cDes_*}}%{\widehat{\cDes}}
\newcommand{\wcdes}{{\cdes_*}}%{\widehat{\cDes}}

\newcommand{\AffDes}{{\operatorname{cDes}}}
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%\newcommand{\AffDes}{\widetilde{{operatorname{Des}}}}}

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\title[Optional shorter title]{Something to do with math}
\title{On cyclic descents for tableaux}

%% 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{%
Ron M.\ Adin%
\thanks{\href{mailto:radin@math.biu.ac.il}{radin@math.biu.ac.il}. Partially supported by by a MISTI MIT-Israel grant.}%
\addressmark{1},
%\and 
Victor Reiner%
\thanks{\href{mailto:reiner@math.umn.edu}{reiner@math.umn.edu}. Partially supported by NSF grant DMS-1601961.}%
\addressmark{2}
\and 
Yuval Roichman%
\thanks{\href{mailto:yuvalr@math.biu.ac.il}{yuvalr@math.biu.ac.il}. Partially supported by by a MISTI MIT-Israel grant.}%
\addressmark{3}
}

\address{%
\addressmark{1}Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel\\
\addressmark{2}School of Mathematics, University of Minnesota,
Minneapolis, MN 55455, USA\\
\addressmark{3}Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel}
%\author{Ron M.\ Adin}
%\address{Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel}
%\email{radin@math.biu.ac.il}
%\author{Victor Reiner}
%\address{School of Mathematics, University of Minnesota,
%Minneapolis, MN 55455, USA}
%\email{reiner@math.umn.edu}
%\author{Yuval Roichman}
%\address{Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel}
%\email{yuvalr@math.biu.ac.il}

%\thanks{First and third authors partially supported by a MISTI MIT-Israel grant.
%Second author partially supported by NSF grant DMS-1601961.}

%% put the date of submission here
\received{November 14, 2017}

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\revised{March 29, 2018}

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\abstract{%
The notion of descent set, for permutations as well as for standard Young tableaux (SYT), is classical.
Cellini introduced a natural notion of {\em cyclic descent set} for permutations, and Rhoades introduced such a notion for SYT --- but only for rectangular shapes.
In this work we define {\em cyclic extensions} of descent sets in a general context, 
and prove existence and essential uniqueness for SYT of almost all shapes. 
The proof applies nonnegativity properties of Postnikov's toric Schur polynomials, providing a new interpretation of certain Gromov--Witten invariants.
}

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%\resume{\lipsum[2]}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{descent, cyclic descent, standard Young tableau,
ribbon Schur function, Gromov--Witten invariant}

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\usepackage[backend=bibtex]{biblatex}
\addbibresource{60-Adin-Reiner-Roichman.bib}
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\begin{document}

\maketitle
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%\tableofcontents


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}

For a permutation $\pi = [\pi_1, \ldots, \pi_n]$ 
in the symmetric group $\symm_n$ on $n$ letters, one defines its {\em descent set} as
\[
\Des(\pi) := \{1 \le i \le n-1 \,:\, \pi_i > \pi_{i+1} \}
\quad \subseteq [n-1],
\]
where $[m]:=\{1,2,\ldots,m\}$.  For example, $\Des([2,1,4,5,3])=\{1,4\}$. On the other hand, its
{\em cyclic descent set} was defined by Cellini~\cite{Cellini} as
\begin{equation}\label{e.cellini}
\cDes(\pi) := \{1 \leq i \leq n \,:\, \pi_i > \pi_{i+1} \}
\quad \subseteq [n],
\end{equation}
with the convention $\pi_{n+1}:=\pi_1$.  For example, $\cDes([2,1,4,5,3])=\{1,4,5\}$.
This cyclic descent set was further studied by Dilks, Petersen, Stembridge~\cite{DPS} and others.
It has several important properties.
Consider the two $\ZZ$-actions, on $\symm_n$ and on the power set of $[n]$, in which the generator $p$ of $\ZZ$ acts by 
\[
\begin{array}{rcl}
[\pi_1,\pi_2,\ldots,\pi_{n-1},\pi_n]&\overset{p}{\longmapsto}&
[\pi_n,\pi_1,\pi_2,\ldots,\pi_{n-1}], \\
\{i_1,\ldots,i_k\}&\overset{p}{\longmapsto}&\{i_1+1,\ldots,i_k+1\} \bmod{n}.
\end{array}
\]
Then, for every permutation $\pi$, one has the following three properties:
\begin{eqnarray}
\label{restriction-property}
&\cDes(\pi) \cap [n-1] = \Des(\pi)&\quad \text{(extension)} \\
\label{equivariance-property}
&\cDes(p(\pi))  = p(\cDes(\pi)) &\quad \text{(equivariance)} \\
\label{non-Escher-property}
&\varnothing \subsetneq \cDes(\pi) \subsetneq [n] &\quad \text{(non-Escher)} 
\end{eqnarray}
The term {\em non-Escher} refers to M.\ C.\ Escher's drawing ``Ascending and Descending'', which paradoxically depicts the impossible cases 
$\cDes(\pi)=\varnothing$ and $\cDes(\pi)=[n]$.

\medskip
There is also an established notion of descent set for a
{\em standard Young tableau} $T$ of a skew shape $\lambda/\mu$:
\[
\Des(T) := \{1 \le i \le n-1 \,:\,
            i+1 \text{ appears in a lower row of }T\text{ than }i\}
\quad \subseteq [n-1].
\]
For example, the following standard Young tableau $T$ of shape $\lambda/\mu=(4,3,2)/(1,1)$ 
has $\Des(T)=\{2,3,5\}$:
\[
\ytableausetup{boxsize=\mysize}
\begin{ytableau}
\none& 1 & 2 & 7 \\
\none& 3& 5 \\
   4 &6 \\
\end{ytableau}
\]
For the special case of standard Young tableaux $T$ of {\em rectangular} shapes,
Rhoades \cite[Lemma 3.3]{Rhoades} introduced a notion of 
{\em cyclic descent set} $\cDes(T)$, having the same properties
\eqref{restriction-property}, \eqref{equivariance-property} and \eqref{non-Escher-property}
with respect to the $\ZZ$-action in which the generator $p$ acts on tableaux via 
Sch\"utzenberger's {\em jeu-de-taquin promotion} operator. 
A similar concept of $\cDes(T)$ and accompanying action $p$ was 
introduced for two-row shapes and certain other skew shapes 
%(see Subsection~\ref{known-examples-subsection} for the list) 
in~\cite{AER, ER1}, 
and used there to answer Schur positivity questions. 

\medskip

Our first main result is a necessary and sufficient condition for the existence of a cyclic extension $\cDes$ of the descent map $\Des$ on the set $\SYT(\lambda/\mu)$ of standard Young tableaux of shape $\lambda/\mu$,
with an accompanying $\ZZ$-action on $\SYT(\lambda/\mu)$ via an operator $p$,
satisfying properties 
\eqref{restriction-property}, \eqref{equivariance-property} and \eqref{non-Escher-property}.
%this notion of cyclic extension is formalized in Section~\ref{sec:definitions} below.
In this story, a special role is played by the skew shapes known as
{\em ribbons} (connected skew shapes containing no $2 \times 2$ rectangle),
and in particular {\em hooks} (straight ribbon shapes, namely $\lambda=(n-k,1^k)$ for $k=0,1,\ldots,n-1$).
Early versions of~\cite{AER} and~\cite{ER1} conjectured the following result. 
%proven in Section~\ref{sec:conj1-proof}.

\begin{theorem}\label{conj1}
Let $\lambda/\mu$ be a skew shape.
The descent map $\Des$ on $\SYT(\lambda/\mu)$ has a cyclic extension $(\cDes,p)$ if and only if $\lambda/\mu$ is not a connected ribbon.
Furthermore, for all $J \subseteq [n]$, all such cyclic extensions share the same cardinalities $\#\cDes^{-1}(J)$.
\end{theorem}

Our strategy for proving Theorem~\ref{conj1} is inspired by a result of Gessel \cite[Theorem 7]{Gessel} that we recall here.
For a subset $J=\{j_1  < \ldots < j_t\} \subseteq [n-1]$, the composition (of $n$)
\begin{equation}\label{composition-of-set}
\comp{n}{J}:= (j_1,j_2-j_1,j_3-j_2,\ldots,j_t-j_{t-1},n-j_t)
\end{equation}
defines a {\em connected ribbon} having the entries of $\comp{n}{J}$ as row lengths, and thus an associated {\em (skew) ribbon Schur function} 
\begin{equation}\label{ribbon-skew-as-alternating-sum}
s_{\comp{n}{J}}
:= \sum_{\varnothing \subseteq I \subseteq J}
(-1)^{\#(J \setminus I)} h_{\comp{n}{I}}
\end{equation}
with the following property: %(ref ??):  
for any skew shape $\lambda/\mu$, the %descent 
map $\Des : \SYT(\lambda/\mu) \rightarrow 2^{[n-1]}$ 
has fiber sizes given by
\begin{equation}\label{Gessel-ribbon-descent-fact}
\# \Des^{-1}(J) =
\langle s_{\lambda/\mu}, s_{\comp{n}{J}} \rangle
\qquad (\forall J \subseteq [n-1]),
\end{equation}
where $\langle -,-\rangle$ is the usual inner product on symmetric functions. 

By analogy, for a subset $\varnothing \ne J = \{ j_1 < j_2 < \ldots < j_t \}
\subseteq [n]$
we define the corresponding {\em cyclic composition} of $n$  as
\begin{equation}\label{cyclic-composition-of-set}
\ccomp{n}{J} := (j_2-j_1, \ldots, j_t - j_{t-1}, j_1 + n - j_t),
\end{equation}
with $\ccomp{n}{J} := (n)$ when $J = \{j_1\}$;
note that $\ccomp{n}{\varnothing}$ is not defined.
The corresponding {\em affine (or cyclic) ribbon Schur function} is
then defined as
\begin{equation}\label{affine-ribbon-Schur-function}
\cs_{\ccomp{n}{J}} := \sum_{\varnothing \neq I \subseteq J}
(-1)^{\#(J \setminus I)} h_{\ccomp{n}{I}}.
\end{equation}
We then collect enough properties of %$\cs_{\ccomp{n}{J}}$ 
this function
to show that there must exist 
a map $\cDes: \SYT(\lambda/\mu) \rightarrow 2^{[n]}$
and a $\ZZ$-action $p$ on $\SYT(\lambda/\mu)$,
as in Theorem~\ref{conj1}, 
such that fiber sizes are given by
\begin{equation}\label{cDes-fiber-sizes-as-inner-product}
\# \cDes^{-1}(J) =
\langle s_{\lambda/\mu}, \cs_{\ccomp{n}{J}} \rangle 
\qquad (\forall\ \varnothing \subsetneq J \subsetneq [n]).
\end{equation}
See Corollary~\ref{cor:fiber-sizes-as-inner-product} below.
The nonnegativity of this inner product when $\lambda/\mu$ is not a connected ribbon ultimately relies on relating
$\cs_{\ccomp{n}{J}}$ to a special case of Postnikov's {\it toric Schur polynomials},
with their interpretation in terms of {\em Gromov--Witten invariants}
for Grassmannians \cite{Postnikov}.

\medskip
We also compare the distribution of $\cDes$ on $\SYT(\lambda)$
to the distribution of $\cDes$ on $\symm_n$.
Recall \cite[Theorem 3.1.1 and Section 5.6 Ex. 22(a)]{Sagan} that the {\it Robinson--Schensted correspondence} is a bijection
between $\symm_n$ and the set of all pairs of standard Young tableaux 
of the same shape (and size $n$), %$\lambda$ partitioning $n$, 
having the property that if $w \mapsto (P,Q)$ then $\Des(w)=\Des(Q)$.
Consequently
\[
\sum_{w \in \symm_n} \ttt^{\Des(w)}
= \sum_{ \lambda \vdash n } f^\lambda \sum_{T \in \SYT(\lambda)} \ttt^{\Des(T)}.
\]
Here $\ttt^S:=\prod_{i \in S}t_i$ for $S \subseteq \{1,2,\ldots\}$,
$\lambda \vdash n$ means that $\lambda$ is a partition of $n$,
and $f^\lambda:=\#\SYT(\lambda)$.
Note that Theorem~\ref{conj1} implies that any {\em non-hook} shape $\lambda$, as well as any disconnected skew shape $\lambda/\mu$, 
will have
$\sum_{T \in \SYT(\lambda/\mu)} \ttt^{\cDes(T)}$ well-defined and
independent of the choice of cyclic extension $(\cDes,p)$ for $\Des$ on $\SYT(\lambda/\mu)$.
We then have the following second main result.

\begin{theorem}\label{conj2}
%Let $\lambda/\mu$ be a skew shape which is not a connected ribbon.
%Then any cyclic extension $(\cDes,p)$ of the descent map $\Des$ on $\SYT(\lambda)$ satisfies
For any $n \ge 2$
\[
\sum_{w \in \symm_n} \ttt^{\cDes(w)}
= \sum_{\substack{\text{non-hook}\\ \lambda \vdash n}}
f^\lambda \sum_{T \in \SYT(\lambda)} \ttt^{\cDes(T)}
\quad + \quad \sum_{k=1}^{n-1} \binom{n-2}{k-1} 
 \sum_{T \in \SYT((1^k)\oplus (n-k))} \ttt^{\cDes(T)},
\]
where $\cDes$ is defined on $\symm_n$ by Cellini's formula \eqref{e.cellini} and on standard Young tableaux (of the relevant shapes) as in Theorem~\ref{conj1}.
\end{theorem}
\noindent
The {\em direct sum} operation $\lambda \oplus \mu$ in the last summation
denotes a skew shape having the diagram of $\lambda$ strictly southwest
of the diagram of $\mu$, with no rows or columns in common.  For example, when
$(n,k)=(7,2)$, 
\[
(1^k)\oplus (n-k)
\quad = \quad (1^2) \oplus (5)
\quad = \quad
\ytableausetup{boxsize=\mysizesmall}
\ydiagram{1,1}
\,\,\, \oplus \,\,\,
\ydiagram{5}
\quad = \quad
\ydiagram{1+5,1,1}
\]

The current work is an extended abstract, stating the definitions and main results together with some indication of the proofs. For a complete exposition, including all the relevant proofs and additional material, see the full paper version \cite{ARR}.


%\section*{Acknowledgments}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{%Cyclic descents:  
Definition, examples, and basic properties}
\label{sec:definitions}


%\subsection{Definition}

Let us begin by formalizing the concept of a cyclic extension.
Recall the bijection $p: 2^{[n]} \longrightarrow 2^{[n]}$ induced by the cyclic shift $i \mapsto i + 1 \pmod n$, for all $i \in [n]$.

\begin{definition}
\label{def:cDes}
Let $\TTT$ be a finite set. A {\em descent map} is any map
$\Des: \TTT \longrightarrow 2^{[n-1]}$. 
A {\em cyclic extension} of $\Des$ is
a pair $(\cDes,p)$, where 
$\cDes: \TTT \longrightarrow 2^{[n]}$ is a map 
and $p: \TTT \longrightarrow \TTT$ is a bijection,
satisfying the following axioms:  for all $T$ in  $\TTT$,
\[
\begin{array}{rl}
\text{(extension)}   & \cDes(T) \cap [n-1] = \Des(T),\\
\text{(equivariance)}& \cDes(p(T))  = p(\cDes(T)),\\
\text{(non-Escher)}  & \varnothing \subsetneq \cDes(T) \subsetneq [n].\\
\end{array}
\]
\end{definition}

The non-Escher axiom will be important for the uniqueness of the cyclic extension. 
%see Subsection~\ref{subsec:Escher} for a discussion of the consequences of omitting this assumption.

%\bigskip

%%%%%%
%\subsection{Known examples}\label{known-examples-subsection}

\begin{example}
Given any (strict) {\em composition} $\alpha$ of $n$,
that is, an ordered sequence of positive integers $\alpha=(\alpha_1,\ldots,\alpha_t)$
with $\sum_i \alpha_i=n$, define the associated {\em horizontal strip} skew shape
\[
\alpha^\oplus :=(\alpha_1) \oplus (\alpha_2) \oplus \cdots \oplus (\alpha_t)
\]
whose rows, from southwest to northeast, have sizes $\alpha_1,\ldots,\alpha_t$. 
For $T$ in $\TTT=\SYT(\alpha^\oplus)$ we define
\[
\cDes(T):=\{1\le i\le n \,:\, i+1 \text{ is in a lower row than } i\},
\]
where $n+1$ is interpreted as $1$,
as well as a bijection $p: \SYT(\alpha^\oplus) \rightarrow \SYT(\alpha^\oplus)$
which first replaces each entry $j$ of $T$ by $j+1 \pmod n$ and then
re-orders each row to make it left-to-right increasing.
One can check that this $(\cDes,p)$ is a cyclic extension of $\Des$,
with $t \ge 2$ required for the non-Escher property.

This generalizes the case of $(\cDes,p)$ on  $\symm_n$, since for $\alpha=(1^n)=(1,1,\ldots,1)$ one has a bijection $\symm_n \rightarrow \SYT(\alpha^\oplus)$ which sends a permutation $w$ to the tableau whose entries are $w^{-1}(1),\ldots,w^{-1}(n)$ read from southwest to northeast. 
This bijection maps Cellini's cyclic extension $(\cDes,p)$ on $\symm_n$ to
the one on $\SYT(\alpha^\oplus)$ for $\alpha=(1,1,\ldots,1)$ defined above\footnote{This cyclic descent map can further be %naturally 
generalized  to {\em strips}, which are the disconnected shapes with each connected component consisting of either one row or one column; see  \cite{AER}.}.
\end{example}

Cyclic extensions of the descent map on $\TTT=\SYT(\lambda)$ have been given previously for several families of shapes: %In particular: cyclic descent extensions were first introduced, for rectangular shapes, by Rhoades
for {\em rectangular shapes} see~\cite[Lemma 3.3]{Rhoades}; for {\em flag shapes},  namely $(k,k,1^{n-2k})$ with $1\le k\le n/2$, see~\cite{Pechenik, DPS2}; for {\em two-row shapes} $(k,ell)$ %see~\cite{AER};
and {\em hook plus internal corner} shapes, namely $(n-2-k,2,1^k)$ with $0 \le k \le n-4$, see~\cite{AER}; for skew shapes of type  $\lambda \oplus (1)$, where
 $\lambda\vdash n-1$ is any nonempty partition, see~\cite{ER1}.
The last case was, in fact, our original motivation, and the question of existence of a cyclic extension of $\Des$ on $\SYT(\lambda/\mu)$ appears there as \cite[Problem 5.5]{ER1}.


%%%%%%%%%%%%%%%%%%%
%\subsection{Existence and uniqueness of cyclic extensions}

\begin{lemma}\label{t.cyclic_extension}
Fix a set $\TTT$ and a map $\Des: \TTT \to 2^{[n-1]}$.

\begin{enumerate}
\item[$(i)$] 
Any cyclic extension $(\cDes,p)$ of $\Des$ has fiber sizes 
$m(J) := \#\cDes^{-1}(J)$
(for $J \subseteq [n]$) satisfying
\begin{itemize}
\item[$(a)$]
$m(J) \ge 0$ for all $J$, with $m(\varnothing)=m([n])=0$;
\item[$(b)$]
$m(J)=m(p(J))$ for all $J$; and
\item[$(c)$]
$m(J) + m(J \sqcup \{n\}) = \Des^{-1}(J)$ for all $J \subseteq [n-1]$.
\end{itemize}

\item[$(ii)$]
Conversely, if 
%$\#\Des^{-1}(J)$ for $J \subseteq [n-1]$ are such that there exist integers 
$(m(J))_{J \subseteq [n]}$ are integers
satisfying conditions (a),(b) and (c) above,
then there exists at least one cyclic extension $(\cDes,p)$ of $\Des$ satisfying $\#\cDes^{-1}(J) = m(J)$ for all $J \subseteq [n]$.

\end{enumerate}

\end{lemma}


%%%%%%%%
%\subsection{Univariate generating functions}
%{The $\cdes$ versus $\des$ generating function}

The definition of a cyclic extension $(\cDes,p)$ of $\Des$ on $\TTT$ leads immediately to a relation between the ordinary generating functions
\[
%\begin{aligned}
\TTT^\cdes(t):=\sum_{T \in \TTT} t^{\#\cDes(T)}, \qquad 
%\]
%and
%\[
\TTT^\des(t):=\sum_{T \in \TTT} t^{\#\Des(T)}.
%\end{aligned}
\]

\begin{lemma}
\label{lem:cdes-des-distribution-relation}
For any cyclic extension $(\cDes,p)$ of $\Des$ on $\TTT$ 
one has
\[
\frac{n \TTT^{\des}(t)}{(1-t)^{n+1}}
= \frac{d}{dt} \left[ 
\frac{\TTT^{\cdes}(t)}{(1-t)^{n}}\right].
\]
\end{lemma}

This lemma completely determines $\TTT^{\cdes}(q)$ in terms of $\TTT^{\des}(q)$,
since the non-Escher condition implies that the polynomial $\TTT^{\cdes}(t)$ has no constant term in $t$, and similarly for the formal power series $\TTT^{\cdes}(t)/(1-t)^n$.

%\begin{proof}
%Since $p$ and each of its powers $p^k$ are bijective, one has 
%$$
%n \TTT^{\des}(t) = \sum_{T \in \TTT} \sum_{k=0}^{n-1} t^{\des(p^k T)}.
%$$
%However, equivariance implies that if $\cdes(T)=c$
%then the ordered list $(T,pT,p^2T,\ldots,p^{n-1}T)$ contains
%\begin{itemize}
%\item[$\bullet$]
%exactly $c$ entries which have $n$ in their cyclic descent set,
%and hence have $\cdes(T)-1$ descents, and
%\item[$\bullet$]
%the remaining $n-c$ elements have $\cdes(T)$ descents.
%\end{itemize}
%Therefore the right side above can be rewritten
%$$
%\sum_{T \in \TTT} \left( \cdes(T) t^{\cdes(T)-1} + (n-\cdes(T)) t^{\cdes(T)} \right)
%= n \TTT^\cdes(t) + (1-t) \frac{d}{dt} \TTT^\cdes(t).
%$$
%It is not hard to check that this is equivalent to the assertion of the lemma.
%\end{proof}


%\begin{remark}
%\label{non-Escher-implies-no-constant-term}
%This lemma completely determines $\TTT^{\cdes}(q)$ in terms of $\TTT^{\des}(q)$,
%since the non-Escher condition implies that the polynomial $\TTT^{\cdes}(t)$ has no constant term in $t$, and similarly for the formal power series $\TTT^{\cdes}(t)/(1-t)^n$.
%\end{remark}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Ribbon Schur functions: affine, cylindric and toric}
\label{sec:Postnikov}

Recall from the introduction that our proof strategy for Theorem~\ref{conj1} involves the introduction of a family of new symmetric functions, which we call {\em affine} (or {\em cyclic}) {\em ribbon Schur functions}.  
We recall here their definition and develop some of their properties, using standard terminology and properties of symmetric functions.
%, as in Macdonald \cite{Macdonald}, Sagan \cite{Sagan}, or Stanley \cite{EC2}.




%%%%%%%
\subsection{Affine ribbon Schur functions}
\label{sec:affine-ribbon-schurs}

%We now introduce {\em cyclic} or {\em affine} analogues of the previous composition and ribbon concepts.  

\begin{definition}
\label{defn_aff}
To each nonempty subset 
$\varnothing \ne J = \{ j_1 < j_2 < \ldots < j_t \} \subseteq [n]$ 
associate (as in \eqref{cyclic-composition-of-set} above) a {\em cyclic composition} of $n$,
\[
\ccomp{n}{J} := (j_2-j_1, \ldots, j_t - j_{t-1}, j_1 + n - j_t).
\]
In particular, $\ccomp{n}{\{j_1\}} := (n)$ while $\ccomp{n}{\varnothing}$ is undefined.
The corresponding {\em affine (or cyclic) ribbon Schur function} is defined (as in \eqref{affine-ribbon-Schur-function} above) by
\[
\cs_{\ccomp{n}{J}} := \sum_{\varnothing \neq I \subseteq J}
(-1)^{\#(J \setminus I)} h_{\ccomp{n}{I}}.
\]
%Define also
%\[
%\cs_{\ccomp{n}{\varnothing}} := 0.
%\]
\end{definition}

\begin{example}
For $n=9$ and $J=\{2,6\}$ the ordinary ribbon Schur function is
\[
\begin{aligned}
s_{\comp{9}{\{2,6\}}}
&= h_{\comp{9}{\{2,6\}}} - h_{\comp{9}{\{6\}}} - h_{\comp{9}{\{2\}}} 
   + h_{\comp{9}{\varnothing}}\\
&= h_{(2,4,3)} - h_{(6,3)}-h_{(2,7)} + h_{(9)}.
\end{aligned}
\]
By contrast, the corresponding {\em affine} ribbon Schur function is
\[
\begin{aligned}
\cs_{\ccomp{9}{\{2,6\}}}
&= h_{\ccomp{9}{\{2,6\}}} -h_{\ccomp{9}{\{6\}}} -h_{\ccomp{9}{\{2\}}} \\
 &= h_{(4,5)} - h_{(9)}-h_{(9)}
= h_{(4,5)} - 2h_{(9)}.
 \end{aligned}
\]
\end{example}

Our proof strategy for Theorem~\ref{conj1} involves
showing that, whenever $\lambda/\mu$ is a skew shape of size $n$ which is not a connected ribbon, the integers
$m(J):=\langle s_{\lambda/\mu}, \cs_{\ccomp{n}{J}} \rangle$
satisfy conditions (a),(b) and (c) of Lemma~\ref{t.cyclic_extension}.
In fact, the nonnegativity condition (a) is the most subtle; 
conditions (b) and (c) follow from the following two propositions.

\begin{proposition}[Equivariance]
\label{cyclic-invariance-of-cyclic-compositions}
For each nonempty subset $\varnothing \ne J \subseteq [n]$, the cyclic composition $\ccomp{n}{p(J)}$ is a cyclic shift of $\ccomp{n}{J}$,
and consequently 
\[
\cs_{\ccomp{n}{p(J)}}=\cs_{\ccomp{n}{J}}.
\]  
\end{proposition}
%\begin{proof}
%Both assertions follow immediately from the definitions.
%%\footnote{\color{red} There may be something to prove here, regarding the invariance of $\cs_\alpha$ under a cyclic shift of $\alpha$.}
%\end{proof}

\begin{proposition}[Extension]\label{t.cor_sum}
For each nonempty $\varnothing \ne J \subseteq [n-1]$,
\[
\cs_{\ccomp{n}{J}}+\cs_{\ccomp{n}{J\sqcup\{n\}}}=s_{\comp{n}{J}}.
\]
%This also holds for $J = \varnothing$, if $\cs_{\ccomp{n}{\varnothing}}$ is interpreted as 0.
\end{proposition}

%\begin{proof}
%For $J = \varnothing$ this holds since
%\[
%\cs_{\ccomp{n}{\{n\}}} 
%= s_{\comp{n}{\varnothing}}
%= h_{(n)}.
%\]
%Assume that $J \ne \varnothing$.
%By definition,
%\[
%\cs_{\ccomp{n}{J}} + \cs_{\ccomp{n}{J\sqcup\{n\}}}
%= \sum_{\varnothing \neq I \subseteq J} 
% (-1)^{\#(J \setminus I)} h_{\ccomp{n}{I}}
%+  \sum_{\varnothing \neq I \subseteq J \sqcup \{n\}}
%(-1)^{\#((J \sqcup \{n\}) \setminus I)} h_{\ccomp{n}{I}}.
%\]
%Each subset $\varnothing \ne I \subseteq J$ contributes 
%an $h_{\ccomp{n}{I}}$ to each of the two sums,
%but with opposite signs, so they cancel each other.  
%The remaining terms (in the second sum)
%correspond to subsets $I$ which contain $n$, 
%and can be written as 
%$I = I' \sqcup \{n\}$ for $I' \subseteq J$:
%\[
%\cs_{\ccomp{n}{J}} + \cs_{\cc{n}{J\sqcup\{n\}}}
%= \sum_{\{n\} \subseteq I \subseteq J \sqcup \{n\}} 
%(-1)^{\#((J \sqcup \{n\}) \setminus I)} h_{\ccomp{n}{I}}
%= \sum_{I' \subseteq J} 
%(-1)^{\#(J \setminus I')} h_{\ccomp{n}{I' \sqcup \{n\}}}.
%\]
%The cyclic composition $\ccomp{n}{I' \sqcup \{n\}}$ is a (cyclic) rearrangement of the ordinary composition $\comp{n}{I'}$. Thus 
%%$h_{\cc(I' )}=h_{\co(\hat{I})}$, and hence one can rewrite this as
%\[
%\cs_{\ccomp{n}{J}} + \cs_{\ccomp{n}{J\sqcup\{n\}}}
%= \sum_{I' \subseteq J} (-1)^{\#(J \setminus I')} h_{\comp{n}{I'}}
%= s_{\comp{n}{J}}. %\qedhere
%\]
%\end{proof}

The subtle nonnegativity condition (a) in Lemma~\ref{t.cyclic_extension} will be derived
from the following result, which is the main goal of this section.

\begin{theorem}\label{conj_a3}
For every $J \subseteq [n]$ and non-hook
partition $\lambda\vdash n$, %one has
\[
\langle \cs_{\ccomp{n}{J}}, s_\lambda \rangle \ge 0.
\]
\end{theorem}


%%%%%%%
\subsection{Cylindric ribbon shapes and cylindric Schur functions}
%{Cylindric and toric ribbon shapes}

The key to Theorem~\ref{conj_a3} is a
relation between the affine ribbon symmetric function
$\cs_{\ccomp{n}{J}}$, defined above, and a special case of 
Postnikov's cylindric Schur functions \cite{Postnikov}
which was introduced implicitly already by 
Gessel and Krattenthaler \cite{GesselKrattenthaler} 
and studied further by McNamara \cite{McNamara}.


%In the current subsection we recall the relevant definitions and results regarding cylindric shapes and cylindric Schur functions, in the special case (cylindric ribbons) that we need. 
%We start by recalling a special case of Postnikov's {\em cylindric shapes} $\lambda/d/\mu$.

\begin{defn}
[{Cf.\ \cite[Section 3]{Postnikov} and \cite[Definition 3.4]{McNamara}}]
\label{def:cylindric-ribbon-shape}
For each subset 
$\varnothing \ne J = \{j_1 < \ldots < j_t\} \subseteq [n]$
define a {\em cylindric ribbon shape} $C_J = \lambda/1/\lambda$ in one of the following equivalent ways:
\begin{itemize}
\item (Postnikov)
The partition
\[
\lambda:= (n - t, j_t - j_1 - (t-1), \ldots, j_3 - j_1 - 2, j_2 - j_1 - 1)
\]
fits inside a $t \times (n-t)$ rectangle, and may be viewed as a lattice path connecting the southwestern and northeastern corners of this rectangle.
Repeat this path, periodically, to obtain an infinite path.
The cylindric ribbon shape $\lambda/1/\lambda$ is the infinite 
ribbon contained between this infinite path and its shift by one step eastward (equivalently, southward).

\item
Consider the cyclic composition 
\[
\ccomp{n}{J}=(j_2-j_1,j_3-j_2,\ldots,j_1+n-j_t).
\]
The cylindric ribbon shape $C_J$ is the infinite ribbon whose sequence of row lengths (from southwest to northeast) is the sequence of parts of this composition, repeated periodically.

\item
Consider the cyclic composition $\ccomp{n}{J}$ above, and let $R_J$ be the corresponding (finite) ribbon shape. Its
northwest boundary is given by the partition $\lambda$ above, except that the first part should be $n-t+1$ rather than $n-t$.
Denote by $a$ (respectively, $b$) the extreme southwestern (respectively, northeastern) square of this ribbon shape.
The cylindric ribbon shape $C_J = \lambda/1/\lambda$ is an infinite 
ribbon made up of copies $(R_i)_{i \in \ZZ}$ of the ribbon $R_J$, placed in the plane such that 
square $a$ of $R_{i+1}$ is immediately north of square $b$ of $R_i$, for all $i \in \ZZ$.
%\begin{comment}
%\item
%Since the $t \times (n+1-t)$ rectangle has one "extra" column compared to
%the $t \times (n-t)$ rectangle, wrap around the $(n+1-t)^{th}$ (rightmost)
%column and {\it glue it on top of} the $1^{st}$ (leftmost) column, giving $\lambda/1/\lambda$.
%\end{comment}
\end{itemize}
\end{defn}


%Recall now, from \cite{Postnikov}, the definition of a {\em toric} shape (here -- in the ribbon case).
\begin{definition}[{\cite[Definition 3.2 and Lemma 3.3]{Postnikov}}]
\label{def:toric_shape}
A cylindric ribbon shape $C_J = \lambda/1/\lambda$ is called {\em toric} if (each) one of the following equivalent conditions holds:
\begin{enumerate}
\item
Each row of $C_J$ has length at most $n-t$.
\item
Each column of $C_J$ has length at most $t$.
\item
The first and last columns of $R_J$ have no squares in the same row. 
\end{enumerate}
\end{definition}


\begin{definition}[{\cite[Section 5]{Postnikov}}]
\label{def:cylindric-Schur}
Let $\lambda/1/\lambda$ be a cylindric ribbon shape. Define the corresponding {\em cylindric Schur function} by
\[
s_{\lambda/1/\lambda}(x_1,\ldots) := \sum_T {\xx}^T,
\]
where summation is over all {\em semistandard cylindric tableaux} $T$ filling the shape $\lambda/1/\lambda$ with entries in $\{1,2,\ldots\}$. This means that the entries of $T$ are ``$(t,n)$-periodic'' (see \cite[Figure 4]{Postnikov}), weakly increasing from left to right in rows and strictly increasing from top to bottom in columns.
\end{definition}




%%%%%%%%
\subsection{Affine vs.\ cylindric ribbon Schur functions}

Our next result shows that our affine ribbon Schur functions 
(Definition~\ref{sec:affine-ribbon-schurs}) are almost the same as Postnikov's cylindric ribbon Schur functions (Definition \ref{def:cylindric-Schur}).
To state it, recall the {\em power sum} symmetric function defined by
$p_n := x_1^n + x_2^n + \ldots$.


\begin{proposition}
\label{toric-zigzag-conjecture}
For $\varnothing \ne J \subseteq [n]$, with associated cylindric ribbon shape $C_J = \lambda/1/\lambda$, one has
\[
s_{\lambda/1/\lambda} = \cs_{\ccomp{n}{J}} + (-1)^{\#J} p_n.
\]
\end{proposition}

%\begin{proof}
%As explained in \eqref{every-h-is-a-skew-schur}, one has
%\begin{equation}
%\label{cyclic-h-tableau-sum}
%h_{\ccomp{n}{J}} = \sum_T {\xx}^T
%\end{equation}
%summing over all semistandard tableaux $T$ 
%filling the horizontal strip $\ccomp{n}{J}^\oplus$.
%For $i=1,2,\ldots,t$ consider the $i^{th}$ row (from the bottom) of this horizontal strip, 
%and label its leftmost and rightmost entries by $x_i$ and $y_i$, respectively.
%For Example~\ref{example:cylindric-ribbon-shape} with $n=9$, $J=\{1,4,5,8\}$ and $\ccomp{n}{J}=(3,1,3,2)$, the horizontal strip is:
%%\begin{comment}
%%\[
%%\ytableausetup{boxsize=\mysizehuge}
%%\begin{ytableau}
%% \none & \none & \none & \none & \none & 
%% \none & \none & x_4   & y_4   \\
%% \none & \none & \none & \none & x_3   & 
%%       &  y_3  & \none & \none \\
%% \none & \none & \none & \mbox{\tiny ${x_2 \!\!=\!\! y_2}$} 
%%% \tiny\begin{matrix}x_2\\ \Vert \\y_2\end{matrix} \small
%% & \none & \none & \none   & \none   & \none \\
%% x_1  &       & y_1   & \none & \none & 
%% \none& \none & \none & \none
%%\end{ytableau}
%%\]
%%\end{comment}
%\[
%\ytableausetup{boxsize=\mysize}
%\begin{ytableau}
% \none & \none & \none & \none & \none & 
% \none & \none & x_4   & y_4   \\
% \none & \none & \none & \none & x_3   & 
%       &  y_3  & \none & \none \\
% \none & \none & \none & \bullet \\
% x_1  &       & y_1   & \none & \none & 
% \none& \none & \none & \none
%\end{ytableau}
%\qquad (\bullet = x_2 = y_2)
%\]
%Comparing this to the corresponding (finite) ribbon
%%\begin{comment}
%%\[
%%R_J = \,
%%\ytableausetup{boxsize=\mysizehuge}
%%\begin{ytableau}
%% \none & \none & 
%% \none & \none & x_4   & y_4   \\
%% \none & \none & x_3   & 
%%       &  y_3  & \none & \none \\
%% \none & \none & \mbox{\tiny ${x_2 \!\!=\!\! y_2}$} 
%%% \tiny\begin{matrix}x_2\\ \Vert \\y_2\end{matrix} \small
%% & \none & \none & \none   & \none   & \none \\
%% x_1  &       & y_1   & \none & \none & 
%% \none& \none & \none & \none
%%\end{ytableau}
%%\]
%%\end{comment}
%\[
%R_J = \,
%\ytableausetup{boxsize=\mysize}
%\begin{ytableau}
% \none & \none & 
% \none & \none & x_4   & y_4   \\
% \none & \none & x_3   & 
%       &  y_3  & \none & \none \\
% \none & \none & \bullet \\
% x_1  &       & y_1   & \none & \none & 
% \none& \none & \none & \none
%\end{ytableau}
%\qquad (\bullet = x_2 = y_2)
%\]
%we see that $s_{\lambda/1/\lambda}$ is equal to the same sum as in \eqref{cyclic-h-tableau-sum}, but only over those
%$T$ which satisfy the strict inequalities 
%\[
%x_i < y_{i-1} \qquad (1 \le i \le t),
%\]
%where subscripts are interpreted modulo $t$ so that the first inequality is $x_1 < y_0 = y_t$.
%
%Define, for each $T$ appearing in \eqref{cyclic-h-tableau-sum},
%its {\em violation set} 
%\[
%V(T) := \{i \,:\, y_{i-1} \le  x_i\} \subseteq [t].
%\]
%Inclusion-exclusion gives
%\[
%s_{\lambda/1/\lambda}
%= \sum_{T\,:\, V(T) = \varnothing} \xx^T
%= \sum_{A \subseteq [t]} (-1)^{\#A} \sum_{T \,:\, V(T) \supseteq A} \xx^T.
%\]
%A violation at $i$ means that $y_{i-1} \le x_i$, 
%so that row $i$ (from the bottom) may be juxtaposed 
%at the end of row $i-1$, keeping $T$ semistandard . 
%Thus, for each $A \subseteq [t]$ other than $A = [t]$,
%the rightmost sum is over all semistandard tableaux filling
%the horizontal strip $\ccomp{n}{I}$ for 
%$I := \{j_i \,;\, i \in [t] \setminus A\} \ne \varnothing$,
%and hence the sum is $h_{\ccomp{n}{I}}$.  
%For $A=[t]$ (i.e., $I = \varnothing$),
%the sum is over the {\em constant} fillings $T$, 
%yielding $x_1^n + x_2^n + \ldots = p_n$.
%Therefore
%\[
%s_{\lambda/1/\lambda}
%= (-1)^{\#J} p_n
%+ \sum_{\varnothing \neq I \subseteq J} 
%(-1)^{\# (J \setminus I)} h_{\ccomp{n}{I}}
%= (-1)^{\#J} p_n + \cs_{\ccomp{n}{J}}. %\qedhere
%\]
%\end{proof}

For an algebraic consequence of the toric property of a shape, consider the specialization $x_{t+1} = x_{t+2} = \ldots = 0$.
%\begin{proposition}\label{t:non-toric-vanishes}{\rm \cite[Lemma 5.2]{Postnikov}}
%The cylindric (toric) Schur polynomial $s_{\lambda/1/\lambda}(x_1, \ldots, x_t)$ is nonzero if and only if the shape $\lambda/1/\lambda$ is toric.
%\end{proposition}

%Together with Lemma~\ref{t:toric-conditions}, this implies
%\begin{corollary}\label{t:hook-vanishes}
%For $\lambda = ((n-t)^i, 0^{t-i})$  with $1 \le i \le t$,
%\[
%s_{\lambda/1/\lambda}(x_1, \ldots, x_t) = 0.
%\]
%\end{corollary}
%In fact, it is not difficult to prove Corollary~\ref{t:hook-vanishes} directly from Definition~\ref{def:cylindric-Schur}, since the shape $C_J$ has a column of length $t+1$ which cannot be filled by distinct numbers in $[t]$.

\begin{proposition}[{Cf.\ McNamara \cite[Lemma 5.3 and Proposition
5.5]{McNamara}}]\label{t.Gromov-Witten-corollary}
Fix $1 \le t \le n$. Then
\begin{equation}
\label{e.cyclic-ribbon-expansion}
\cs_{\ccomp{n}{[t]}}=\sum_{k=0}^{t-1} (-1)^{t-1-k} s_{(n-k,1^k)}
\end{equation}
and, for each nonempty subset $\varnothing \ne J \subseteq [n]$ with $\#J = t$, there exist nonnegative integers $c_{J,\nu}$ such that
\begin{equation}
\label{e.Gromov-Witten-expansion}
\cs_{\ccomp{n}{J}}
= \cs_{\ccomp{n}{[t]}} 
+ \sum_{\text{non-hook } \nu \vdash n} c_{J,\nu} s_\nu.
\end{equation}
\end{proposition}

%\begin{proof}
%We claim that it suffices to prove both assertions after specializing them to 
%the finite variable set $\{x_1,\ldots,x_t\}$, namely
%letting $x_{t+1} = x_{t+2}= \ldots = 0$.  
%Indeed, this specialization annihilates each Schur function $s_{\nu}$ with $\ell(\nu) > t$, whereas the surviving $\{s_\nu(x_1,\ldots,x_t)\}_{\ell(\nu) \leq t}$ form a basis for the symmetric polynomials in $x_1,\ldots,x_t$; see
%\cite[Chap. I, (3.2)]{Macdonald}.
%%sending each of the basis elements $\{s_\nu\}$ for
%%all symmetric functions either to $0$ if $\ell(\nu) > t$, or to the basis elements
%%$\{s_\nu(x_1,\ldots,x_t)\}_{\ell(\nu) \leq t}$ for the symmetric polynomials in $x_1,\ldots,x_t$.  
%Thus our claim will follow once we check that, for $t = \#J$,
%$\cs_{\ccomp{n}{J}}$ always lies in the linear span of the Schur functions $s_\nu$ with $\ell(\nu) \leq t$.
%According to Definition~\ref{defn_aff},
%$\cs_{\ccomp{n}{J}}$ is an alternating sum of $h_{\ccomp{n}{I}}$ 
%with each $I$ of size $ 1 \le \#I \le t$, so that
%$\alpha = \ccomp{n}{I}$ is a composition of $n$ with at most $t$ parts,
%and our claim follows from {\em Young's rule} 
%\cite[Corollary 7.12.4]{EC2}: 
%\[
%h_\alpha=\sum_{\nu} K_{\nu,\alpha} s_\nu,
%\]
%where $K_{\nu,\alpha}$ is the number of semistandard tableaux of shape $\nu$ having $\alpha_j$ occurrences of the entry $j$ (for each $j$), which is zero if $\ell(\nu) > t$.
%
%Now, letting $x_{t+1} = x_{t+2}= \ldots = 0$, equation~\eqref{e.power_sum} specializes to
%\[
%p_n(x_1, \ldots, x_t) = 
%\sum_{k = 0}^{t-1} (-1)^{k} s_{(n-k,1^k)}(x_1, \ldots, x_t),
%\]
%since $s_{(n-k,1^k)}(x_1,\ldots,x_t)=0$ for $k \ge t$.
%We can thus rephrase the two assertions of our proposition as follows:
%for each $J \subseteq [n]$ of size $t$ there exist $c_{J,\nu} \ge 0$ such that 
%\[
%\cs_{\ccomp{n}{J}}(x_1,\ldots,x_t)
%= (-1)^{t-1} p_n(x_1,\ldots,x_t) 
%+ \sum_{\text{non-hook } \nu} c_{J,\nu} s_\nu(x_1,\ldots,x_t),
%\]
%and if $J=[t]$ then all $c_{J,\nu}=0$.
%Using Proposition~\ref{toric-zigzag-conjecture}, this is equivalent 
%to the assertion that there exist $c_{J,\nu} \geq 0$ such that 
%\[
%s_{\lambda/1/\lambda}(x_1,\ldots,x_t) 
%= \sum_{\text{non-hook } \nu} c_{J,\nu}  s_\nu(x_1, \ldots, x_t),
%\]
%and that for $J=[t]$ they all vanish, 
%i.e.,  $s_{\lambda/1/\lambda}(x_1,\ldots,x_t)=0$.
%%\begin{comment}
%%Indeed, for $J=[t]$ one has $\ccomp{n}{J} = (1, \ldots, 1, n-t+1)$
%%and $\lambda = (n-t, 0, \ldots, 0)$, 
%%which supports no cylindric tableaux with entries 
%%in $\{1,2,\ldots,t\}$, due to the requirement of strict
%%inequalities $a_1 < a_2 < \cdots < a_t < a_{t+1}$ here:
%%\begin{equation}
%%\ytableausetup{boxsize=\mysizelarge}
%%\begin{ytableau}
%%\none    &\none    & \none  & \none & \none&  \none  &\none&\vdots \\
%%\none    &\none    & \none  & \none & \none&  \none  &\none&a_t\\
%%\none    &a_1 &            &             &         &            &        &a_{t+1}    \\
%%\none    &a_2   & \none & \none  & \none &\none&\none&\none \\
%%\none    &\vdots   & \none & \none  & \none   & \none&\none &\none \\
%% \none    &a_t    & \none  & \none & \none&  \none  &\none&\none\\
%%\cdots    & a_{t+1}    & \none  & \none & \none&  \none  &\none&\none
%%\end{ytableau}
%%\end{equation}
%%\end{comment}
%Indeed, for $J = [t]$ one has $\lambda = (n-t, 0^{t-1})$ and therefore, by Corollary~\ref{t:hook-vanishes}, $s_{\lambda/1/\lambda}(x_1,\ldots,x_t)=0$.
%On the other hand, for any $J \subseteq [t]$, 
%the inequalities $c_{J,\nu} \ge 0$ follow from
%Postnikov's result \cite[Theorem 5.3]{Postnikov} that
%\[
%s_{\lambda/d/\mu}(x_1,\ldots,x_t)
%= \sum_{\nu \subseteq [t] \times [n-t]} C^{\lambda,d}_{\mu,\nu} s_\nu(x_1, \ldots, x_t)
%\]
%where the sum is over all shapes $\nu$ contained in a $t \times (n-t)$ rectangle, and $C^{\lambda,d}_{\mu,\nu}$ are
%{\em Gromov--Witten invariants} appearing as structure constants in the quantum cohomology of Grassmannians, and known to be nonnegative. Since no hook shape of size $n$ is contained in a $t \times (n-t)$ rectangle, the sum is over non-hook shapes only, for which $c_{J,\nu} = C^{\lambda,1}_{\lambda,\nu} \ge 0$, and the result follows.
%\end{proof}

Theorem~\ref{conj_a3} is clearly a consequence of Proposition~\ref{t.Gromov-Witten-corollary}, since for any nonempty subset $J \subseteq [n]$ and any non-hook partition $\nu \vdash n$
\[
\langle \cs_{\ccomp{n}{J}}, s_\nu \rangle
= \langle s_{\lambda/1/\lambda}, s_\nu \rangle
= C^{\lambda,1}_{\lambda,\nu} \ge 0.
\]

\medskip
It is worth noting that the Gromov--Witten invariants $C^{\lambda,d}_{\mu,\nu}$ have several interpretations, in addition to the one given in the proof of Proposition~\ref{t.Gromov-Witten-corollary}:
\begin{itemize}
\item[$\bullet$] 
They count certain {\em puzzles}, as conjectured by Knutson
and proved by Buch, Kresch, Purbhoo, and Tamvakis~\cite{BKPT}.
\item[$\bullet$] 
They have algebraic interpretations involving Morse's {\em $k$-Schur functions}, and in the {\em Verlinde fusion algebra};
see, e.g., the %background 
discussion by Morse and Schilling \cite[Section 1.4]{MorseSchilling}.
\item[$\bullet$] 
Pawlowski has proved \cite[Theorem 7.8]{Pawlowski} a conjecture of Postnikov \cite[Conjecture 9.1]{Postnikov}, asserting that that $s_{\lambda/d/\mu}$ is the Frobenius characteristic for the Specht module of the toric shape ${\lambda/d/\mu}$, so that $C^{\lambda,d}_{\mu,\nu}$ are its irreducible expansion coefficients.
\end{itemize}

In the special case where $d=1$ and $\lambda=\mu$, the shape $\lambda/1/\lambda = C_J$ corresponds to some $J \subseteq [n]$,
and then, for a (non-hook) shape $\nu$, one can regard Theorem~\ref{conj1} as yielding another interpretation: %of $C^{\lambda,1}_{\lambda,\nu}$:
\[
C^{\lambda,1}_{\lambda,\nu} = \langle \cs_{\ccomp{n}{J}}, s_\nu \rangle
= \#\{ T \in \SYT(\nu): \cDes(T)=J \},
\]
where $(\cDes,p)$ is any cyclic extension of $\Des$ on $\SYT(\nu)$.


%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Hook multiplicities in skew Schur functions}
%\label{sec:lemmas}
%
%We will now compute, for future use, the multiplicities of hook Schur functions in the Schur expansion of a skew Schur function.
%Recall that a {\em ribbon} is a connected skew shape which 
%does not contain a $2 \times 2$ square. A {\em generalized ribbon} is a skew shape all of whose connected components are ribbons. 
%The {\em height} of a skew shape is the number of its nonempty rows. 
%
%\begin{lemma}\label{lemma_prel2}
%Fix a hook shape $(n-k,1^k)$ with $0 \le k \le n-1$.
%Then, for a skew shape $\lambda/\mu$ with $n$ cells and $m$ connected components,
%\[
%\langle s_{\lambda/\mu}, s_{(n-k,1^k)} \rangle =
%\begin{cases}
%\binom{m-1}{h-k-1}, & \text{if }   \lambda/\mu \text{ is a generalized ribbon of height } k+1 \le h \le k+m;\\
%0, & \text{otherwise.}
%\end{cases}
%\]
%\end{lemma}
%\begin{proof}
%By the Littlewood-Richardson rule ~\cite[Theorems A1.3.3 and A1.3.8]{EC2}), 
%the LHS is the number of %size of the set $\LLL$ of all 
%{\em semi-standard} (column-strict) Young tableaux $T$ of shape $\lambda/\mu$,
%filled with $n-k$ copies of $1$ and exactly one copy of each of 
%$2, 3,\ldots, k+1$, 
%such that the {\em reading word} of $T$ (the concatenation of all its rows, bottom to top) has the sequence $k+1,k,\ldots,2,1$ as a subword.
%
%There are no such tableaux $T$ if $\lambda/\mu$ contains a $2 \times 2$ square,
%since this square must be filled with 
%%\begin{comment}
%%$
%%\ytableausetup{boxsize=\mysizesmall}
%%\begin{ytableau}
%%1 & 1  \\
%%a& b 
%%\end{ytableau}
%%$
%%where $2 \leq a < b \leq k+1$,  preventing $b,b-1,\ldots,a+1, a$ 
%%from proceeding northeast.
%%\end{comment}
%\[
%\ytableausetup{boxsize=\mysizesmall}
%\begin{ytableau}
%a & b  \\
%c & d 
%\end{ytableau}
%\]
%where $a < c \le d$, implying $1 < c < d$ and violating the subword condition.
%
%Thus $\lambda/\mu$ must be a generalized ribbon, say with $m$ connected components and height $h$.
%The subword condition implies that each row of $T$ can have at most one entry which is not $1$, and that these entries are $k+1, k, \ldots, 2$, from bottom to top. The semi-standard property of $T$ implies that each such entry is in the easternmost square in its row, and the ribbon shapes of the components imply that the easternmost entry in a row can be a $1$ only for the top row of a component.
%
%Summing up, let $I$ be the set of all connected components that have a $1$ at their northeastern corner. Then $I$ completely determines the tableau $T$: all the top rows corresponding to $I$ end with a $1$, all the other rows in $T$ end with a non-$1$, namely with $2, \ldots, k, k+1$ in increasing order from top to bottom, and all other entries are $1$. The set $I$ must include the top (northeastern) component, by the subword property, but is otherwise free. The number of possible such $I$ (i.e., the number of tableaux $T$) is $\binom{m-1}{\#I-1}$.
%The observations that $1 \le \#I \le m$ and $h = k + \#I$ (by row counting) complete the proof.
%\end{proof}
%
%\begin{corollary}\label{lemma_prel4}
%For a generalized ribbon $\lambda/\mu$ of size $n$ and height $h$, with $m \geq 2$ components, there exist nonnegative coefficients $c_\nu \geq 0$ such that
%\[
%\begin{aligned}
%s_{\lambda/\mu} 
%&= \sum_{\text{non-hook }\nu\vdash n} c_\nu s_\nu 
%+ \sum_{k=h-m}^{h-1} \binom{m-1}{h-1-k} s_{(n-k, 1^k)} \\
%&= \sum_{\text{non-hook }\nu\vdash n} c_\nu s_\nu 
%+ \sum_{k=h-m+1}^{h-1} \binom{m-2}{h-1-k} s_{(1^k) \oplus (n-k)}
%\end{aligned}
%\]
%\end{corollary}
%
%\begin{proof}
%The first equality follows from Lemma~\ref{lemma_prel2}, since the skew Schur function $s_{\lambda/\nu}$ is a nonnegative linear combination of Schur functions.
%The second equality follows, then, from the equation
%%For non-hook $\nu$, one has $c_\nu = \langle s_{\lambda/\mu}, s_\nu \rangle \geq 0$.
%%Using a consequence of the Pieri formula (or the Littlewood-Richardson rule),
%\begin{equation}\label{e.sum_of_hooks}
%s_{(1^k) \oplus (n-k)}=s_{(n-k+1,1^{k-1})}+s_{(n-k,1^k)}
%\qquad (1 \le k \le n-1).
%\end{equation}
%%This, together with Lemma~\ref{lemma_prel2} and the Pascal triangle recurrence, gives the rest of the expansion.
%This equation, in turn, follows easily from the definition of a Schur function as a sum over semi-standard tableaux $T$, distinguishing the cases $T_a \ge T_b$ and $T_a < T_b$, where $a$ is the first (westernmost) square of the row shape $(n-k)$ and $b$ is the top square of the column shape $(1^k)$.  
%\end{proof}
%
%%Lemma~\ref{lemma_prel2} also has consequences for $\cs_{\ccomp{n}{J}}$.
%%First, it lets one be more precise in \eqref{e.Gromov-Witten-expansion}: one has
%%$$
%%\begin{aligned}
%%\cs_{\ccomp{n}{J}}
%%&= \cs_{\ccomp{n}{[t]}} + \sum_{\substack{\text{non-hooks}\\\nu}} c_{J,\nu} s_\nu  
%%= \sum_{k=0}^{t-1} (-1)^{t-1-k} s_{(n-k,1^k)}
%%+ \sum_{\substack{\text{non-hooks}\\\nu}} c_{J,\nu} s_\nu
%%\end{aligned}
%%$$
%%which is equivalent to the following assertion.
%An immediate consequence of Proposition~\ref{t.Gromov-Witten-corollary} is the following.
%
%\begin{corollary}
%\label{affine-ribbon-hook-pairing} 
%For $\varnothing \ne J \subseteq [n]$ with $t:=\#J$ and $0 \le k \le n-1$,
%\[
%\langle \cs_{\ccomp{n}{J}}, s_{(n-k,1^k)} \rangle =\begin{cases}
%(-1)^{t-1-k} & \text{ if } 0 \le k \le t-1;\\
%0 & \text{ otherwise. }
%\end{cases}
%\]
%\end{corollary}
%%\begin{proof}
%%By Lemma~\ref{lemma_prel2},
%%%Young's rule \eqref{Youngs-rule},
%%if $\alpha$ has $\ell(\alpha)$ parts then 
%%$ \langle h_\alpha ,
%%s_{(n-k,1^k)} 
%%\rangle = %K_{(n-i,1^i),\alpha} =
%%\binom{\ell(\alpha)-1}{k}$. Consequently,
%%if $t\le k$ then
%%$\langle \cs_{\ccomp{n}{J}}, s_{(n-k,1^k)}
%%\rangle=0$. 
%%Otherwise
%%$$
%%\begin{aligned}
%%\langle \cs_{\ccomp{n}{J}}, s_{(n-k,1^k)} \rangle
%%&=\sum_{ \varnothing \neq I \subseteq J} (-1)^{t-\#I}  \langle h_{\ccomp{n}{I}}, s_{(n-k,1^k)} \rangle
%%=(-1)^t \sum_{ \varnothing \neq I \subseteq J} (-1)^{\#I} \binom{\#I-1}{k}\\
%%&= (-1)^t \sum_{ i=1}^t (-1)^i \binom{t}{i}   \binom{i-1}{k} =(-1)^{t-1-k}
%%\end{aligned}
%%$$
%%where we have suppressed in the last step a binomial identity.
%%
%%%
%%% Here's a non-hypergeometric proof that Vic did, but probably not worth including
%%\begin{comment}
%%
%%$$
%%\begin{aligned}
%%(-1)^t \sum_{ k=1}^t (-1)^k \binom{t}{k}   \binom{k-1}{i}
%%= (-1)^t t \sum_{ k=1}^t (-1)^k \frac{1}{k}\binom{t-1}{k-1}   \binom{k-1}{i}\\
%%= (-1)^{t-1} t \left[ \int \sum_{ k=1}^t \binom{t-1}{k-1}   \binom{k-1}{i} (-x)^{k-1} dx \right]_{x=1}\\
%%= (-1)^{t-1} [z^i] t \left[ \int\sum_{i=0}^{k-1} \sum_{ k=0}^{t-1}  \binom{t-1}{k-1}   \binom{k-1}{i} z^i (-x)^{k-1} dx \right]_{x=1}
%%\end{aligned}
%%$$
%%\end{comment}
%%\end{proof}
%
%Together with equation \eqref{e.sum_of_hooks}, this implies the following.
%\begin{corollary}
%\label{cor:crucial-hook-fact}
%For $\varnothing \ne J \subseteq [n]$ with $t=\#J$ and $1 \le k \le n$,
%\[
%\langle \cs_{\ccomp{n}{J}}, s_{(1^k) \oplus (n-k)}  \rangle =
%\begin{cases}
%1 & \text{ if } k=t,\\
%0 & \text{ otherwise.}
%\end{cases}
%\]
%\end{corollary}
%%\begin{proof}
%%Recall that by \eqref{e.sum_of_hooks},
%%\[
%%s_{(1^k) \oplus (n-k)} =s_{(1^k,n-k)}+s_{(1^{k-1},n-k+1)}.
%%\] 
%%Hence Lemma~\ref{affine-ribbon-hook-pairing} implies
%%\[
%%\begin{aligned}
%%&\langle \cs_{\ccomp{n}{J}}, s_{(1^k) \oplus (n-k)} \rangle\\
%%&=\langle \cs_{\ccomp{n}{J}}, s_{(1^k,n-k)} \rangle
%%  + \langle \cs_{\ccomp{n}{J}}, s_{(1^{k-1},n-k+1)} \rangle 
%%=\begin{cases} 
%%0+0=0, & \text{if } k > t,\\
%%0+(-1)^{t-1-(k-1)}=+1, & \text{if } k=t,\\
%%(-1)^{t-1-k}+(-1)^{t-1-(k-1)}=0 & \text{if } k<t. %\qedhere\\
%%\end{cases} 
%%\end{aligned}
%%\]
%%\end{proof}
%%\todo{Due to changes in the proof of Theorem~\ref{conj1} in Section~\ref{sec:conj1-proof}, Corollary~\ref{cor:crucial-hook-fact} is no longer used in the paper. Should we keep it for its own interest?}





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Proof of Theorem~\ref{conj2}}
%\label{sec:conj2-proof}
%
%Recall the statement of the theorem.
%\vskip.1in
%\noindent
%{\bf Theorem~\ref{conj2}.}
%{\em
%Let $\lambda/\mu$ be a skew shape which is not a connected ribbon.
%Then any cyclic extension $(\cDes,p)$ of the descent map $\Des$ on $\SYT(\lambda)$ satisfies
%\[
%\sum_{w \in \symm_n} \ttt^{\cDes(w)}
%= \sum_{\substack{\text{non-hook}\\ \lambda \vdash n}}
%f^\lambda \sum_{T \in \SYT(\lambda)} \ttt^{\cDes(T)}
%\quad + \quad \sum_{k=1}^{n-1} \binom{n-2}{k-1} 
% \sum_{T \in \SYT((1^k)\oplus (n-k))} \ttt^{\cDes(T)}
%\]
%}
%
%\noindent
%This is actually the special case $\alpha=(1,1,\ldots,1)=(1^n)$ of the following more general statement, which requires a bit more notation.
%Recall that, for a partition $\lambda \vdash n$ and a composition $\alpha=(\alpha_1,\ldots,\alpha_m)$ of $n$, the {\it Kostka number} 
%\[
%K_{\lambda, \alpha} :=\langle s_\lambda, h_\alpha\rangle
%\] 
%counts the column-strict tableaux $T$ of shape $\lambda$ having content $\alpha$, namely $\alpha_i = \#T^{-1}(i)$ ($\forall i$).
%In particular, 
%\[
%K_{\lambda,(1^n)}=f^\lambda=\#SYT(\lambda).
%\]
%Denote, for a tableau $T$ as above, $\xx^T := x_1^{\alpha_1} \cdots x_m^{\alpha_m}$.
%
%\begin{theorem}\label{prop25}
%For every composition $\alpha=(\alpha_1,\ldots,\alpha_m)$  of $n$ with $m \geq 2$,
%\[
%\sum_{T\in \SYT( \alpha^\oplus)} \ttt^{\cDes(T)}
%= \sum_{\substack{\text{non-hook}\\ \lambda \vdash n}}
%  K_{\lambda,\alpha}  \sum\limits_{T\in \SYT(\lambda)}
%   {\ttt}^{\cDes(T)}
%\quad + \quad
%  \sum_{k=1}^{m-1}\binom{m-2}{k-1} 
%      \sum_{T\in \SYT((1^k)\oplus (n-k))} \ttt^{\cDes(T)},
%\]
%\end{theorem}
%
%\begin{proof}[Proof of Theorem~\ref{prop25}.]
%The horizontal strip $\alpha^\oplus$ is a generalized ribbon with $m \ge 2$ components and height $h = t$. By Corollary~\ref{lemma_prel4},
%\[
%s_{\alpha^\oplus} 
%= \sum_{\text{non-hook } \nu \vdash n} c_\nu s_\nu
%+ \sum_{k=1}^{m-1} \binom{m-2}{m-1-k} s_{(1^k) \oplus (n-k)}
%\]
%for some nonnegative coefficients $c_\nu$.
%Since $s_{\alpha^\oplus} = h_\alpha$,
%\[
%c_\nu = \langle s_{\alpha^\oplus}, s_\nu \rangle 
%= \langle h_\alpha, s_\nu \rangle = K_{\nu,\alpha}
%\]
%and therefore
%\begin{equation}\label{ribbon-schur-Kostka-expansion}
%s_{\alpha^\oplus} 
%= \sum_{\text{non-hook } \nu \vdash n} K_{\nu,\alpha} s_\nu
%+ \sum_{k=1}^{m-1} \binom{m-2}{k-1} s_{(1^k) \oplus (n-k)}.
%\end{equation}
%Consider, for each $J \subseteq [n]$, the coefficient of $\ttt^J$ in the LHS of the statement of the theorem.  By \eqref{cDes-fiber-sizes-as-inner-product} (see the proof of Theorem~\ref{conj1}) it is equal to
%\[
%\#\{T\in \SYT(\alpha^\oplus):\ \cDes(T)=J\}
%= \langle s_{\alpha^\oplus}, \cs_{\ccomp{n}{J}} \rangle,
%\]
%and by \eqref{ribbon-schur-Kostka-expansion} this is equal to
%\[
%\begin{aligned}
%\langle s_{\alpha^\oplus}, \cs_{\ccomp{n}{J}} \rangle
%&= \sum_{\text{non-hook } \nu \vdash n} K_{\nu,\alpha} \langle s_\nu, \cs_{\ccomp{n}{J}} \rangle 
%+ \sum_{k=1}^{m-1} \binom{m-2}{k-1}
%  \langle s_{(1^k) \oplus (n-k)}, \cs_{\ccomp{n}{J}} \rangle \\
%&= \sum_{\text{non-hook } \nu \vdash n} K_{\nu,\alpha} \cdot \#\{T\in \SYT(\nu) \,:\, \cDes(T)=J\} \\
%&+ \sum_{k=1}^{m-1} \binom{m-2}{k-1} \cdot \#\{T\in \SYT((1^k) \oplus (n-k)) \,:\, \cDes(T)=J\},
%\end{aligned}
%\]
%which is exactly the coefficient of $\ttt^J$ in the RHS of the statement of the theorem.
%\end{proof}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Remarks and questions}
\label{sec:remarks-and-questions}

We close with several remarks and questions raised by this work.  
%In some cases, proofs have been suppressed for the sake of brevity.

%\subsection{Cyclic Eulerian distributions}
%\label{sec:symmetric-group}

\bigskip

%\subsection{Cyclic descent generating functions}

\noindent{\bf 4.1. Bijective proofs and cyclic sieving.} 
Detailed proofs of Theorems~\ref{conj1} and~\ref{conj2} may be found in the full version paper~\cite[]{ARR}.
Our proof of the existence of $(\cDes,p)$ in Theorem~\ref{conj1}, whose strategy was sketched above,
is indirect and involves arbitrary choices.

\begin{problem}
\label{natural-maps-problem}
Find a natural, explicit map $\cDes$ and cyclic action $p$ on $\SYT(\lambda/\mu)$ as in Theorem~\ref{conj1}.
\end{problem}

%\noindent
%Theorem~\ref{conj2} is proved in Section~\ref{sec:conj2-proof}. 
%The distribution $\sum_{w \in \symm_n} \ttt^{\cDes(w)}$ is studied further in Section~\ref{sec:symmetric-group}.

\begin{problem}
\label{Schensted-problem}
Find a Robinson--Schensted style bijective proof of Theorem~\ref{conj2}.
\end{problem}
 
Let us state explicitly a consequence of the proof of Theorem~\ref{conj1}.

\begin{corollary}\label{cor:fiber-sizes-as-inner-product}
Let $\lambda/\mu$ be a skew shape of size $n$ which is not a connected ribbon. For any $J \subseteq [n]$ and every cyclic extension $\cDes$ of the usual descent map on $\SYT(\lambda/\mu)$, 
the fiber size
\[
\# \cDes^{-1}(J) =
\langle s_{\lambda/\mu}, \cs_{\ccomp{n}{J}} \rangle .
%	\qquad (\forall J \subseteq [n]).
\]
\end{corollary}

For $J=\{j_1,\cdots,j_t\}\subseteq [n]$ let $-J:=\{-j_1,\dots,-j_t\}$ (interpreted modulo $n$).
%One deduces that 

\begin{corollary}\label{cor:fiber-sizes-involution}
Let $\lambda/\mu$ be a skew shape of size $n$ which is not a connected ribbon. For any $J \subseteq [n]$ and any cyclic extension $\cDes$ of the usual descent map on $\SYT(\lambda/\mu)$, 
the fiber size
\[
\# \cDes^{-1}(J) =\# \cDes^{-1}(-J). 
\]
\end{corollary}

\begin{problem}
For a solution of Problem~\ref{natural-maps-problem}, 
find an %{\em cyclic evacuation} 
involution
on $\SYT(\lambda/\mu)$ which sends the cyclic descent set to its negative.
\end{problem}

%\noindent{\bf 4.4. Cyclic sieving phenomenon.}

\begin{problem}
\label{CSP-problem}
For non-ribbon shapes $\lambda/\mu$, 
can one choose the operator $p$ in Theorem~\ref{conj1}
and a polynomial $X(q)$ %to replace $f^\lambda(q)$
to obtain a cyclic sieving phenomenon (CSP) ?
\end{problem}

This problem was solved by Rhoades \cite{Rhoades}
for rectangular shapes and by Pechenik~\cite{Pechenik} for %flag 
shapes $(k,k,1^{n-2k})$.
%; see also~\cite[\S 2.8]{Iraci}. This result may be used to define an explicit cyclic descent extension for SYT of this shape.
%\medskip
 %Furthermore,
Recalling from~\cite{ER1} the cyclic descent extension for $\SYT(\lambda \oplus (1))$, 
%where $\lambda\vdash n-1$,
Corollary~\ref{cor:fiber-sizes-as-inner-product} in the current paper has been applied in~\cite{ARS} to obtain a refined CSP on SYT of these skew shapes.
%Finally, an elegant refinement of Rhoades' result
%on rectangular shapes was very recently conjectured, and proved for two-row shapes~\cite{ARS}.

%\bigskip
%
%
%\noindent{\bf 4.2 Cyclic Eulerian distributions.}
%The {\em descent number} is the size of the descent set.
%For any skew shape $\lambda/\mu$ of size $n$ there is a known expression \cite[equation (7.96)]{EC2} for the generating function of the descent number, $\des$, on %standard Young tableaux of shape 
%$\SYT(\lambda/\mu)$:
%\begin{equation}
%\label{skew-tableau-des-distribution}
%\sum_{T \in \SYT(\lambda/\mu)} t^{\des(T)}
%= (1-t)^{n+1} \sum_{m \ge 0} s_{\lambda/\mu}(1^{m+1}) t^m.
%\end{equation}
%Here $s_{\lambda/\mu}(1^m)$ is the specialization of the skew Schur function $s_{\lambda/\mu}(x_1, x_2, \ldots)$ under $x_1 = \ldots = x_m = 1$ and $x_{m+1} = \ldots = 0$.
%An analogous expression for the {\em cyclic descent number} $\cdes$ 
%is a corollary of Lemma~\ref{lem:cdes-des-distribution-relation}.
%
%\begin{corollary}\label{cor:cdes_gf_skew}
%For any skew shape $\lambda/\mu$ of size $n$ which is not a connected ribbon,
%\begin{equation}
%\label{skew-tableau-cdes-distribution}
%\sum_{T \in \SYT(\lambda/\mu)} t^{\cdes(T)}
%= n (1-t)^{n} \sum_{m \ge 1} s_{\lambda/\mu}(1^m) \frac{t^m}{m}.
%\end{equation}
%%In particular, for the skew shape $(1)^{\oplus n}$ this gives
%%\begin{equation}
%%\label{cyclic-Carlitz-formula}
%%\symm_n^{\cdes}(t) 
%%:= \sum_{w \in \symm_n} t^{\cdes(w)}
%%= n (1-t)^n \sum_{m \geq 1} m^{n-1} t^m  
%%= nt\, \symm_{n-1}^\des(t) \qquad (n \ge 2).
%%\end{equation}
%\end{corollary}
%%\begin{proof}
%%For  $\TTT=\SYT(\lambda/\mu)$, 
%%Lemma~\ref{lem:cdes-des-distribution-relation} and Remark~\ref{non-Escher-implies-no-constant-term}
%%show that $\TTT^\cdes(t)$ is determined by
%%\[
%%\frac{d}{dt}
%%\left[
%%\frac{\TTT^\cdes(t) }{(1-t)^n}
%%\right]
%%=
%%\frac{n \TTT^\des(t) }{(1-t)^{n+1}}
%%= n \sum_{m \ge 0} s_{\lambda/\mu}(1^{m+1}) t^m.
%%\]
%%%One can easily check that the RHS of \eqref{skew-tableau-cdes-distribution} satisfies this same relation.
%%This implies \eqref{skew-tableau-cdes-distribution}, which for $\lambda/\mu=(1)^{\oplus n}$ specializes to \eqref{cyclic-Carlitz-formula}.
%%\end{proof}
%
%More results and questions regarding cyclic descent generating functions may be found in~\cite{ARR}.

\bigskip


%%%%%%%%%
%\subsection{Topological interpretation of affine ribbon Schur functions}
%\label{Steinberg-torus-section}

%\noindent{\bf 4.3. Topological interpretation of affine ribbon Schur functions.}
\noindent{\bf 4.2. Topological interpretation of affine ribbon Schur functions.}
The alternating sum definition \eqref{affine-ribbon-Schur-function} of the affine ribbon Schur function 
$\cs_{\ccomp{n}{J}}:= \sum_{\varnothing \neq I \subseteq J}
(-1)^{\#(J \setminus I)} h_{\ccomp{n}{I}}$ has a topological interpretation,
as (the Frobenius image of) a certain virtual Euler characteristic representation of $\symm_n$.
In particular, the special case $t=n$ of \eqref{e.cyclic-ribbon-expansion},
\begin{equation}
\label{Steinberg-torus-Euler-Poincare}
\sum_{\varnothing \neq I \subseteq [n]} (-1)^{n-\#I}h_{\ccomp{n}{I}} = \cs_{\ccomp{n}{[n]}} = \sum_{i=0}^{n-1} (-1)^{n-1-i} s_{(n-i,1^i)},
\end{equation}
is the {\em Euler--Poincar\'e relation} for the (ordinary, non-reduced) homology of the (type $A_{n-1}$) {\em Steinberg torus} considered in \cite{DPS}.


%We first r
Recall 
the well-known homological interpretation \cite[Section 6]{Bjorner},
\cite{Solomon}, \cite[Section 4]{Stanley} of $s_{\comp{n}{J}}$.
%the known topological interpretation for the ribbon skew Schur function in terms of 
The {\em type $A_{n-1}$ Coxeter complex} is 
a simplicial complex $\Delta$ triangulating an $(n-2)$-dimensional sphere.
%, with a simply transitive action of $\symm_n$ on its maximal simplices.
It has a {\em balanced} coloring of its vertices by $[n-1]$:  each maximal simplex has exactly one vertex of each color.
Furthermore, for each $J \subseteq [n-1]$, the group $\symm_n$ acts transitively on the simplices whose vertices have color set $J$. By the Euler--Poincar\'e relation for $\Delta_J$ 
%but now with $\symm_n$-stabilizers conjugate to the Young subgroup $\symm_{\comp{n}{J}}$ associated to the composition $\comp{n}{J}$.  Thus the permutation representation of $\symm_n$ on simplices of color set $J$ has image, under the {\it Frobenius characteristic map} $\ch$, equal to the symmetric function $h_{\comp{n}{J}}$.  
%This completely describes the action of $\symm_n$ on the simplices of $\Delta$.

%On the homological side, since $\Delta$ triangulates a sphere, it is {\em Cohen--Macaulay}, and hence has only top-dimensional (reduced) homology.
%This Cohen--Macaulay property is also inherited, for each subset $J \subseteq [n-1]$, by the {\em type-selected subcomplex} $\Delta_J$ consisting of the simplices that only use vertices whose colors lie in $J$. 
%The Euler--Poincar\'e relation for $\Delta_J$ says that
%\[
%\sum_{i \geq -1} (-1)^i \ch( \tilde{H}_i(\Delta_J) )
%= \sum_{i \geq -1} (-1)^i \ch( \tilde{C}_i(\Delta_J) )
%\]
%where $\tilde{C}_i, \tilde{H}_i$ are (augmented/reduced) chain
%and homology groups, taken with rational coefficients.
%By the above discussion this gives
%\[
%(-1)^{\#J-1} \ch ( \tilde{H}_{\#J-1}(\Delta_J) )
%=\sum_{I \subseteq J} (-1)^{\#I -1} h_{\comp{n}{I}}
%\]
%Re-writing this last line gives a 
%well-known homological re-interpretation \cite[\S 6]{Bjorner}, \cite{Solomon}, \cite[\S 4]{Stanley} of $s_{\comp{n}{J}}$:
\[
\ch (\tilde{H}_{\#J-1}(\Delta_J))
=\sum_{I \subseteq J} (-1)^{\#(J \setminus I)} h_{\comp{n}{I}}
= s_{\comp{n}{J}},
\]
where $\ch$ is the  {\it Frobenius characteristic map}.

We wish to similarly re-interpret, for $\varnothing \neq J \subseteq [n]$, the affine ribbon skew Schur function $\cs_{\ccomp{n}{J}}$
in terms of the type $A_{n-1}$ case of what Dilks, Petersen and Stembridge \cite{DPS} call the {\em Steinberg torus}.
This is a regular cell complex which we shall denote $\SteinbergTorus$.
It is a {\em Boolean cell complex}: all lower intervals in the partial ordering of cells are Boolean algebras, so that the cells are essentially simplices, but their intersections are not necessarily common faces.
It triangulates an $(n-1)$-dimensional torus, with a simply transitive action of $\symm_n$ on the maximal cells.
It also has a balanced coloring of its vertices by $[n]$, so each maximal simplex has exactly one vertex of each color.
Furthermore, for $\varnothing \neq J \subseteq [n]$, the group $\symm_n$ again acts transitively on the set of all cells whose vertices have color set $J$,
but this time with $\symm_n$-stabilizers conjugate to the Young subgroup $\symm_{\ccomp{n}{J}}$ associated to the {\em cyclic} composition $\ccomp{n}{J}$.  
Thus the permutation representation of $\symm_n$ on the cells of color set $J$ has image, under the Frobenius characteristic map $\ch$, equal to the symmetric function $h_{\ccomp{n}{J}}$.  This
describes the action of $\symm_n$ on the simplices of $\SteinbergTorus$.

On the homological side,
since $\SteinbergTorus$ triangulates a torus, it is not Cohen--Macaulay.
In fact, its (non-reduced) cohomology ring $H^*(\SteinbergTorus)$ 
with rational coefficients is isomorphic to an exterior algebra $\wedge V$, where $V = H^1(\SteinbergTorus)$ carries the irreducible reflection representation of $\symm_n$.  Since
$\wedge^i V$ has Frobenius image $\ch(\wedge^i V)=s_{(n-i,1^i)}$,
this describes the $\symm_n$-action on homology.
An analysis via the Euler--Poincar\'e relation, as before, shows that
\[
\sum_{i \geq 0} (-1)^i \ch( C^i(\SteinbergTorus) ) 
= \sum_{i \geq 0} (-1)^i \ch( H^i(\SteinbergTorus) )
\]
which becomes exactly \eqref{Steinberg-torus-Euler-Poincare} by
the above discussion.  
More generally, for each $\varnothing \neq J \subseteq [n]$, the {\em type-selected subcomplex} $\SteinbergTorus_J$ consisting of the cells that only use vertices whose colors lie in $J$ has Euler--Poincar\'e relation
\[
\sum_{i \geq 0} (-1)^i \ch( C^i(\SteinbergTorus_J) ) 
= \sum_{i \geq 0} (-1)^i \ch( H^i(\SteinbergTorus_J) )
\]
giving the re-interpretation
\[
\cs_{\ccomp{n}{J}}
=\sum_{\varnothing \neq I \subseteq J} (-1)^{\#(J \setminus I)} h_{\ccomp{n}{I}}
=\sum_{i \geq 0} (-1)^{\#J-1 - i} \ch (H^i(\SteinbergTorus_J)).
\]


\acknowledgments{The authors thank Sergi Elizalde, Ira Gessel, Alex Postnikov and Richard Stanley for stimulating discussions, 
and Connor Ahlbach, Anders Bj\"orner, Ale\-ssandro Iraci, Oliver Pechenik, Kyle Petersen, Jim Propp, Brendon Rhoades, Josh Swanson and Michelle Wachs 
for helpful comments.}

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