%% if you are submitting an initial manuscript then you should have submission as an option here
%% if you are submitting a revised manuscript then you should have revision as an option here
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\documentclass[finalversion]{FPSAC2018}
\articlenumber{84}
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\newtheorem{thm}{Theorem}[section]
\newtheorem{lem}[thm]{Lemma}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{ex}[thm]{Example}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{rem}[thm]{Remark}
\newtheorem{conj}[thm]{Conjecture}

\usepackage{lipsum}

%%%% COMMANDS START

\usepackage{wrapfig}
\usepackage{subfigure}
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%
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%
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%%%%% COMMANDS END



%% define your title in the usual way
\title{Product formulas for standard tableaux
  of a family of skew shapes}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author{Alejandro H.
Morales\thanks{\href{mailto:ahmorales@math.umass.edu}{ahmorales@math.umass.edu}. Partially supported by an AMS--Simons travel grant}\addressmark{1}, Igor
Pak\thanks{\href{mailto:pak@math.ucla.edu}{pak@math.ucla.edu}. Partially
supported by the NSF grant DMS:1363193.}\addressmark{2}, and Greta
Panova\thanks{\href{mailto:panova@math.upenn.edu}{panova@math.upenn.edu}.
Partially supported by the NSF grant DMS:1500834.}\addressmark{3}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Department of Mathematics and Statistics,
  UMass,  Amherst, MA \\ \addressmark{2}Department of Mathematics, UCLA,
  Los Angeles, CA \\ \addressmark{3}Department of Mathematics, UPenn,
  Philadelphia, PA; IAS, Princeton, NJ}

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

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

%% put your English abstract here, or comment this out if you don't have one yet
%% please don't use custom commands in your abstract / resume, as these will be displayed online
%% likewise for citations -- please don't use \cite, and instead write out your citation as something like (author year)
\abstract{We give new product formulas for the number of standard Young tableaux
of a six parameter family of skew shapes generalizing a formula by
DeWitt and a formula of Kim and Oh.  These are proved by utilizing symmetries
for evaluations of factorial Schur functions and the Naruse hook
length formula for skew shapes.}

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

%% put your keywords here, or comment this out if you don't have them yet
\keywords{standard tableaux, skew shapes, factorial Schur functions}

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

%% this sample uses biblatex, which (among other things) takes care of URLs in a more flexible way than bibtex
%% but you can use bibtex if you want
\usepackage[backend=bibtex]{biblatex}
\addbibresource{84-Morales-Pak-Panova.bib}
%% note the \printbibliography command at the end of the file which goes with these biblatex commands

\begin{document}

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



\section{Introduction}

\subsection{Foreword}
It is a truth universally acknowledged, that a combinatorial theory
is often judged not by its intrinsic beauty but by the examples and
applications.  Fair or not, this attitude is historically grounded
and generally accepted.  While eternally challenging, this helps
to keep the area lively, widely accessible, and growing in
unexpected directions.


\begin{comment}
There are two notable types of examples and applications one can think of: \ts
\emph{artistic} \ts and \ts \emph{scientific} \ts (cf.~\cite{Gow}).  The former are
unexpected results in the area which are both beautiful and
mysterious.  The fact of their discovery is the main application,
even if they can be later shown by a more direct argument.
%
The latter are results which represent
a definitive progress in the area, unattainable by other means.
To paraphrase Struik, this is ``something to take home'', rather
than to simply admire (see~\cite{Rota}).  While the line is often
blurred, examples of both types are highly desirable, with
the best examples being both artistic and scientific.
\end{comment}


The full version \cite{MPP3} of this extended abstract is a third in a series and continues our study of the
\emph{Naruse hook-length formula}~\eqref{eq:Naruse}, its generalizations and
applications.  In the first paper~\cite{MPP1}, we introduced two
$q$-analogues of the NHLF and gave their (difficult) bijective proofs.
In the second paper \cite{MPP2}, we investigated the special case
of \emph{ribbon hooks}, which were used to obtain two new elementary proofs
of NHLF in full generality, as well as various new mysterious summation and
determinant formulas.

\smallskip


In this extended abstract we exploit the symmetry in a multivariate identity of
evaluations of
factorial Schur functions to give new product formulas for the number of \emph{standard Young tableaux}
of certain skew shapes.


\nin

As an immediate consequence of our results,  we obtain the exact asymptotic formulas which
were unreachable until now (see \cite[Sections 6,8]{MPP3}). Below we illustrate our results one by one, leaving full statements and generalizations
for later.


\subsection{Number of SYT  of skew shape} \label{intro:skewshapesprod}
\emph{Standard Young tableaux} are fundamental objects in enumerative and
algebraic combinatorics and their enumeration is central to the area.  The number  \ts $f^{\lambda} = \bigl|\SYT(\lambda)\bigr|$ \ts
of standard Young tableaux of shape~$\lambda$ of size~$n$,
is given by the classical \emph{hook-length formula}:
\begin{equation}
\label{eq:HLF} \tag{HLF}
f^{\lambda} \, = \, n! \, \prod_{u\in  [\lambda]} \frac{1}{h(u)}\..
\end{equation}
Famously, there is no general product formula for the
number  \ts $f^{\lambda/\mu} = \bigl|\SYT(\lambda/\mu)\bigr|$ \ts  of standard Young tableaux of skew
shape~$\lambda/\mu$.\footnote{In fact, even for small zigzag shapes $\pi=(k+1,k,\ldots,1)/(k-1,k-2,\ldots,1)$, 
the number $f^\pi$ can have large prime divisors.} 
However, such formulas do exist for a few
sporadic families of skew shapes and truncated shapes (see~\cite{AR}).

In this paper we give a six-parameter family of skew shapes (see Figure~\ref{fig:intro_shapes}~(iv)) with product formulas 
for the number of their SYT.  This product formula is given in our
main result: Theorem~\ref{thm:skewprod}.
The three corollaries below showcase the most elegant special cases. We single out
two especially interesting special cases: \ts Corollary~\ref{cor:abcde-shape} due to its 
connection to the \emph{Selberg integral}, 
and Corollary~\ref{cor:abc-shape} due to its connection to the shifted shapes 
and a potential for a bijective proof.  Both special 
cases are known, but their proofs do not generalize in this paper's direction. 

The formulas below are written in terms of {\em superfactorials} $\Phi(n)$,
{\em double superfactorials} $\Lam(n)$,
{\em super doublefactorials} $\Psi(n)$, and {\em shifted super doublefactorials}
$\Psi(n;k)$ defined as:
\begin{align}
\Phi(n)\. &:=\. 1!\cdot 2! \ts \cdots \ts (n-1)!\,,  \qquad \quad
            \.\. \Lam(n)\. := \. (n-2)!(n-4)!\ts \cdots\.,  \label{eq:defsuperfact}\\ 
\Psi(n) & := \. 1!! \cdot 3!! \ts \cdots \ts (2n-3)!!\,, \qquad
          \Psi(n;k)\.  :=\.  (k+1)!!\ts \cdots \ts (k+3)!! \ts \cdots
          \ts (k+2n-3)!! \notag
\end{align}


\smallskip

\begin{cor}[Kim--Oh~\cite{KO}] \label{cor:abcde-shape}
For all \ts $a,b,c,d,e \in \nn$, let $\lambda/\mu$ be the skew
shape in \ts {\rm Figure~\ref{fig:intro_shapes}~(i)}. Then
the number \ts $f^{\lambda/\mu} = \bigl|\SYT(\lambda/\mu)\bigr|$ \ts is equal to
\[
n! \, \. \frac{\Phi(a)\.\Phi(b)\.\Phi(c)\.\Phi(d)\.
\Phi(e)\.\Phi(a+b+c)\.\Phi(c+d+e)\.\Phi(a+b+c+d+e)}{\Phi(a+b)\.
\Phi(d+e)\.\Phi(a+c+d)\.\Phi(b+c+e)\.\Phi(a+b+2c+d+e)}\,,
\]
where  $n=|\la/\mu|=(a+c+e)(b+c+d)-ab-ed$.
\end{cor}

Note that in~\cite[Corollary 4.7]{KO}, the product formula is equivalent, 
but stated differently.  

\smallskip

\begin{cor}[DeWitt~\cite{DeW}] \label{cor:abc-shape}
For all \ts  $a,b,c\in \nn$, let $\lambda/\mu$ be the skew shape
in \ts {\rm Figure~\ref{fig:intro_shapes}~(ii)}.
Then the number  \ts $f^{\lambda/\mu} = \bigl|\SYT(\lambda/\mu)\bigr|$ \ts  is equal to
\[
n! \,  \,  \frac{\Phi(a)\.\Phi(b)\.\Phi(c)\.
\Phi(a+b+c)\ts\cdot\ts \Psi(c)\.\Psi(a+b+c)}{\Phi(a+b)\.\Phi(b+c)
\.\Phi(a+c)\ts\cdot\ts  \Psi(a+c)\.\Psi(b+c)\.\Psi(a+b+2c)}\,,
\]
where   $n=|\la/\mu|=\binom{a+b+2c}{2}-ab$.
\end{cor}

\smallskip

\begin{figure}[hbt]
\begin{center}
\includegraphics{deltaabc_abcde_abcdem_shapes}
\caption{Skew shapes with product formulas for the number of SYT. The
  symbol $m$ in the last shape indicates slope.}
\label{fig:intro_shapes}
\end{center}
\end{figure}

\begin{cor} \label{cor:abcde1-shape}
For all \ts  $a,b,c,d,e\in \nn$, let $\lambda/\mu$ be the skew
shape in \ts {\rm Figure~\ref{fig:intro_shapes}~(iii)}. Then the number
\ts $f^{\lambda/\mu} = \bigl|\SYT(\lambda/\mu)\bigr|$ \ts  is equal to
$$
% \begin{aligned}
%& C(a,b,c,d,e) \, := \,
%\\
 \frac{n! \.\cdot \. \Phi(a) \ts \Phi(b) \ts \Phi(c) \ts \Phi(a+b+c) \ts\cdot\ts
  \Psi(c;d+e)\ts \Psi(a+b+c;d+e) \ts\cdot \ts\Lam(2a+2c) \ts\Lam(2b+2c)}
{\Phi(a+b) \ts\Phi(b+c)  \ts\Phi(a+c)\cdot \Psi(a+c) \ts\Psi(b+c)\ts
    \Psi(a+b+2c;d+e) \cdot \Lam(2a+2c+d) \ts\Lam(2b+2c+e)}\,,
%\end{aligned}
$$
where $\ts n=|\la/\mu|=(a+b+c+e)(b+c+d)+\binom{a+c}{2}+\binom{b+c}{2}-ab-ed$.
\end{cor}

Let us emphasize that the proofs of
corollaries~\ref{cor:abcde-shape}--\ref{cor:abcde1-shape}
are quite technical in nature.  Here is a brief non-technical explanation.  Fundamentally,
the Naruse hook-length formula (NHLF) provides a new way to understand SYT of skew shape,
coming from geometry rather than representation theory.  What we show in this paper
is that the proof of the NHLF has ``hidden symmetries'' which can be turned into
product formulas.  We refer to Section~\ref{sec:skewprod} for
the complete proofs and common generalizations of these results.

% \bigskip


\subsection{Structure of the paper}
We begin with Section~\ref{sec:not} which summarizes both the notation
and gives a brief review of the earlier work.  In the next Section~\ref{sec:multi},
we develop the technology of multivariate formulas including the key identity
(Theorem~\ref{thm:thickstrip}). We use this identity
to prove the product formulas for the number $f^{\la/\mu}$ of SYT of skew shape in
Section~\ref{sec:skewprod},  including generalization of
corollaries~\ref{cor:abcde-shape}--\ref{cor:abcde1-shape}.

\bigskip

\section{Notation and Background} \label{sec:not}



\subsection{Young diagrams and skew shapes} \label{ss:not-yd}
%
Let $\lambda=(\lambda_1,\ldots,\lambda_r),
\mu=(\mu_1,\ldots,\mu_s)$ denote integer partitions of
length $\ell(\lambda)=r$ and $\ell(\mu)=s$. The {\em size} of the partition
is denoted by $|\lambda|$ and $\lambda'$
denotes the {\em conjugate partition} of $\lambda$. We use $[\lambda]$ to
denote the \emph{Young diagram} of the partition $\lambda$. The \emph{hook length}
$h_{\lambda}(i,j) = \la_i - i +\la_j' -j +1$ of a square $u=(i,j)\in [\la]$ is
the number of squares directly to the right and directly below~$u$
in~$[\la]$ including $u$. The \emph{content} of a square $u = (i,j)$
is $c(u)=j-i$.

A {\em skew shape} is the set difference of the Young diagrams of two
partitions $[\mu] \subseteq [\lambda]$ and we denote it by  $\lambda/\mu$. The \emph{staircase} shape is denoted by
$\delta_n=(n-1,n-2,\ldots,2,1)$.

% \footnote{In \cite{MPP3} the condition for slim shapes is  $\lambda_d \geq \mu_1 + d$.}.


\begin{comment}
\subsection{Permutations} \label{sec:perms}
We write permutations of $\{1,2,\ldots,n\}$
as $w=w_1w_2\ldots w_n \in S_n$, where $w_i$ is the
image of~$i$. Given a positive integer~$c$, let $1^c \times w$ denote the
\emph{direct sum permutation}
$$1^c \times w \. := \. 1\ts 2\ts \ldots \ts c \, (c+w_1)\ts (c+w_2)\.\ldots \. (c+w_n)\..$$
Similarly, let $w\otimes 1^c$ denote the \emph{Kronecker product permutation}
of size $\ts c\ts n\ts$ whose permutation
matrix equals the Kronecker
product of the permutation matrix $P_w$ and the identity $I_c$. See
Figure~\ref{fig:1stshape} for an example.

To each permutation $w\in S_n$, we associate the subset of $[n]\times [n]$
given by
$$
D(w)\. = \. \bigl\{(i,w_j) \ts \mid \ts i<j, \ts w_i<w_j\bigr\}\ts.
$$
This set is called the
(Rothe) \emph{diagram} of $w$ and can be viewed as the complement in
$[n]\times [n]$ of the
hooks from the cells $(i,w_i)$ for $i=1,2,\ldots,n$. The size of this set is the length of $w$ and
it uniquely determines $w$. Diagrams
of permutations play the role in the theory of Schubert polynomials
that  partitions play in the theory of symmetric functions. The {\em essential set} of a
permutation $w$ is given by
$$Ess(w) \. = \. \bigl\{(i,j) \in D(w) \, \bigl| \, (i+1,j),
(i,j+1),(i+1,j+1)\not\in D(w)\bigr\}\ts.$$
See Figure~\ref{fig:permdiags}
for an example of a diagram $D(w)$ and $Ess(w)$.

\begin{figure}[hbt]
\subfigure[]{
\includegraphics[scale=0.6]{461532}
\quad
\includegraphics[scale=0.6]{461532_shape}
\quad
\includegraphics[scale=0.6]{461532_shape_skew}
\label{fig:permdiagsvex}
}
\quad\qquad
\subfigure[]{
\includegraphics[scale=0.6]{251634}
\label{fig:permdiags321}
}
\caption{(a) The diagram of the vexillary permutation $w=461532$ (with
  cells in the essential set tiled in red). Up to permuting rows and
  columns it is the diagram of $\mu(w)=4321$; the supershape
  $\lambda(w)=55332$ defined by the essential set; the
  skew shape $\lambda(w)/\mu(w)$. (b) Example of
  correspondence between skew shapes and $321$-avoiding
  permutations for $w=251634$.}
\label{fig:permdiags}
\end{figure}

The diagrams of two families of permutations have very appealing
properties. These families are also
described using the notion of {\em pattern avoidance} of
permutations and play an important role in Schubert
calculus. We refer to \cite[Section 2.1-2]{Man} for details and further
examples.

A permutation is {\bf vexillary}
if $D(w)$ is, up to permuting rows and columns, the Young diagram of a
partition denoted by $\mu=\mu(w)$. Equivalently, these are 2143-avoiding
permutations, i.e.\ there is no sequence $i<j<k<\ell$ such that $w_j<w_i<w_\ell
<w_k$.  Given a vexillary permutation let
$\lambda=\lambda(w)$ be the smallest partition containing the diagram
$D(w)$. This partition is also the union over the $i\times j$
rectangles with NW--SE corners $(1,1)$, $(i,j)$ for each $(i,j) \in
Ess(w)$. We call this
partition the {\bf supershape} of $w$ and note that $\mu(w)\subseteq
\lambda(w)$ (see Figure~\ref{fig:permdiagsvex}). Examples of vexillary permutations are {\bf dominant} permutations
(132-avoiding) and {\bf Grassmannian} permutations (permutations with at
most one descent).


A permutation is {\bf 321}-avoiding if there is no sequence $i<j<k$
such that $w_i>w_j>w_k$. The diagram $D(w)$ of such a permutation is, up
to removing rows and columns of the board not present
in the diagram and flipping columns, the Young diagram of a skew shape
that we denote $\skewsh(w)$. Conversely, every skew shape $\lambda/\mu$ can be
obtained from the diagram of $321$-avoiding permutation~\cite{BJS}.

\begin{thm}[Billey--Jockusch--Stanley \cite{BJS}] \label{thm:shape2perm}
For every skew shape $\lambda/\mu$ with $(n-1)$ diagonals, there is a
$321$-avoiding permutation $w\in S_n$, such that $\skewsh(w)=\lambda/\mu$.
\end{thm}

The construction from \cite{BJS} to prove this theorem is as follows, label the diagonals of
$\lambda/\mu$ from right to left $\sf{d}_1,\sf{d}_2,\ldots$ and label
the cells of $\lambda/\mu$ by the index of the their diagonal. Let $w$
be the permutation whose reduced word is obtained by reading the
labeled cells of the skew shape from left to right top to bottom
(see Figure~\ref{fig:permdiags321}); we denote this
reduced word by $\rw(\lambda/\mu)$. Note that $\rw(\lambda/\mu)$ is the {\em
  lexicographically minimal} among the reduced words of~$w$.

% For more
% details on the properties of both classes of permutations see \cite[\S 2.1-2]{Man}.
\end{comment}


\subsection{Plane partitions}\label{ss:not-PP}

Let $\RPP(a,b,c)$ denote the set of reverse plane
partitions that fit into an $\ts [a\times b \times c] \ts$ box.  Recall the
\emph{MacMahon box formula} for the number of such  (reverse)
plane partitions, which also can be written as follows:
%
\begin{equation} \label{eq:macmahon}
\bigl|\RPP(a,b,c)\bigr| \. = \, \prod_{i=1}^a\prod_{j=1}^{b}\prod_{k=1}^c \.
\frac{i+j+k-1}{i+j+k-2} = \frac{\Phi(a+b+c)\.\Phi(a)\.\Phi(b)\.\Phi(c)}{\Phi(a+b)\.\Phi(b+c)\.\Phi(a+c)}\,.
\end{equation}

\subsection{Factorial Schur functions} \label{sec:factorial_schurs}

The
{\em factorial Schur function} (e.g.,~see \cite{MoS}) is
defined as
\begin{equation} \label{eq:fschur-explicit}
s_{\mu}^{(d)}({\bf x} \mid {\bf a}) := \, \sum_{T} \. \prod_{u \in \mu}
\. \bigl(x_{T(u)} - a_{T(u)+c(u)}\bigr)\ts,
\end{equation}
where \ts ${\bf x} = x_1,\ldots,x_d$ \ts are variables, \ts ${\bf a} =a_1,a_2,\ldots$
\ts are parameters, and where the sum is over semistandard Young tableaux $T$ of shape $\mu$ with
entries in $\{1,\ldots,d\}$. Moreover,  in addition, \ts
$s_{\mu}^{(d)}({\bf x} \,\vert\, {\bf a})$ \ts is symmetric in $x_1,\ldots,x_d$.


\subsection{Excited diagrams}

Let $\lambda/\mu$ be a skew partition and $D$ be a subset of the Young
diagram of $\lambda$. A cell $u=(i,j) \in D$ is called {\em active} if
  $(i+1,j)$, $(i,j+1)$ and $(i+1,j+1)$ are all in
$[\lambda]\setminus D$.  Let $u$ be an
active cell of $D$, define $\alpha_u(D)$ to be the set obtained by
replacing $(i,j)\in D$ by $(i+1,j+1)$. We call this procedure an {\em excited move}.
An {\em excited diagram} of $\lambda/\mu$ is a subdiagram of $\lambda$ obtained
from the Young diagram of~$\mu$ after a sequence of excited moves on active cells.
Let $\ED(\lambda/\mu)$ be the set of excited diagrams
of~$\lambda/\mu$. See Figure~\ref{ex:excited-diagrams} for examples.

\begin{figure}[hbt]
\begin{center}
\includegraphics[scale=0.85]{ex333-21}.
\end{center}
\caption{The excited diagrams of the shapes (i) $333/21$ and (ii) $333/22$.}
\label{ex:excited-diagrams}
\end{figure}


\begin{comment}
\subsection{Flagged tableaux} \label{ss:excited-flagged}

Excited diagrams of $\lambda/\mu$ are
equivalent to certain {\em flagged tableaux} of shape~$\mu$
(see \cite[\S 3]{MPP1} and \cite[\S 6]{Kre}). The number of
excited diagrams is given by a determinant, a
polynomial in the parts of $\lambda$ and~$\mu$ as follows.
Consider the diagonal that passes through cell $(i,\mu_i)$, i.e.\ the
last  cell of row $i$ in $\mu$. Let this diagonal intersect the
boundary of $\lambda$ at a row denoted by
$\ssf^{(\lambda/\mu)}_i$.  Given an excited diagram $D$ in
$\ED(\lambda/\mu)$ each
$(x,y)$ in $[\mu]$ corresponds to a cell $(i_x,j_y)$ in $D$, let
$\varphi(D):=T$ be the tableau of shape $\mu$ with $T_{x,y} =
j_y$.

\begin{prop}[\cite{MPP1}] \label{prop:flagged}
The map $\varphi$ is a bijection between excited diagrams of $\lambda/\mu$
and SSYT of shape $\mu$ with entries in row $i$ at
most $\ssf^{(\lambda/\mu)}_i$. Moreover,
\[
|\ED(\lambda/\mu)| = \det\left[
  \binom{\ssf^{(\lambda/\mu)}_i+\mu_i-i+j-1}{\ssf^{(\lambda/\mu)}_i-1}\right]_{i,j=1}^{\ell(\mu)}.
\]
\end{prop}

When the last part of $\lambda$ is long enough relative to the parts
of $\mu$ the number of excited diagrams is given by a product.  Recall
the notion of slim shapes~$\la/\mu$ defined in Section~\ref{ss:not-yd}.

\begin{cor} \label{cor:EDhkcontent}
Let $\lambda/\mu$ be a slim skew shape, \ts $d = \ell(\lambda)$.  Then
\[
|\ED(\lambda/\mu)|  \, = \, s_{\mu}(1^d) \, = \, \prod_{(i,j) \in [\mu]} \,
\frac{d + j-i}{\mu_i+\mu'_j -i-j+1}\,.
\]
\end{cor}

\begin{proof}
In this case, by Proposition~\ref{prop:flagged}, the excited
diagrams of $\lambda/\mu$ are in bijection with SSYT of shape $\mu$
with entries at most $d$. The number of such SSYT is given by the
hook-content formula for $s_{\mu}(1^{d})$, see e.g.~\cite[Corollary
7.21.4]{EC}.
\end{proof}

Next we give a family of skew shapes that come up in the paper with product formulas for
the number of excited diagrams.

\begin{ex}[thick reverse hook] \label{ex:excited_macmahon}
 For the shape $\lambda/\mu=(b+c)^{a+c}/b^a$: the excited
  diagrams correspond to SSYT of shape $b^a$ with entries at most
  $a+c$. By subtracting $i$ to the elements in row $i$ these SSYT are
  equivalent to RPP that fit into an \ts $[a\times b \times c]\ts $ box.
  Thus, \ts $|\ED(\lambda/\mu)| = |\RPP(a,b,c)|$ \ts is given by the MacMahon box
  formula \eqref{eq:macmahon}.
\end{ex}
\end{comment}


\subsection{Non-intersecting paths}

Excited diagrams of $\lambda/\mu$ are in bijection with families of non-intersecting grid paths $\ga_1,\ldots,\ga_k$ with a fixed set of
start and end points, which depend only on~$\la/\mu$.  A variant of
this was proved
by Kreiman \cite[Sections 5--6]{Kre} (see also \cite[Section 3]{MPP2}).

\smallskip

Formally, given a connected skew shape $\lambda/\mu$, there is unique family of non-intersecting paths
 $\ga^*_1,\ldots,\ga^*_k$ in $\lambda$ with support
$\lambda/\mu$, where each border strip $\ga^*_i$ begins at the southern box
$(a_i,b_i)$ of a column and ends at the eastern box $(c_i,d_i)$ of
a row
\cite[Lemma 5.3]{Kre}.  Let $\NIP(\lambda/\mu)$ be the set of $k$-tuples
$\Gamma:=(\ga_1,\ldots,\ga_k)$ of non-intersecting paths contained in
$[\lambda]$ with $\ga_i:(a_i,b_i)\to (c_i,d_i)$.

\begin{prop}[Kreiman~\cite{Kre}, see also \cite{MPP2}] \label{prop:excited2nips}
Non-intersecting paths in $\NIP(\lambda/\mu)$ are uniquely determined by
their \emph{support}, i.e.\ set of squares.  Moreover, the set of such
supports is exactly the set of complements $[\la]\sm D$ to excited diagrams
$D\in \ED(\lambda/\mu)$.
\end{prop}

\begin{comment}
\begin{ex} \label{ex:444-21}
The complements of excited diagrams in $\ED(444/21)$ correspond to
tuples $(\ga_1,\ga_2)$ of nonintersecting paths in $[444]$ with
$\ga_1=(3,1)\to (1,4)$ and $\ga_2 = (3,3)\to (2,4)$:
\begin{center}
\includegraphics{exexcited2paths}
\end{center}
\end{ex}
\end{comment}





\subsection{The Naruse hook-length formula}
Recall the formula of Naruse for
$f^{\lambda/\mu}$ as a sum of product of hook-lengths (see~\cite{MPP1}).
% devoted to the study of this formula.
% See \cite{Ko} for a bijective proof.

\begin{thm}[NHLF; Naruse \cite{Nar}] \label{thm:IN}
Let $\lambda,\mu$ be partitions, such that $\mu \ssu \la$.  We have:
\begin{equation} \label{eq:Naruse} \tag{NHLF}
f^{\lambda/\mu} \,  = \, n! \, \sum_{D \in \ED(\lambda/\mu)}\,\.\.
 \prod_{u \in [\lambda]\setminus D} \frac{1}{ h(u)}\.\ts\.,
\end{equation}
where the sum is over all excited diagrams $D$ of $\lambda/\mu$.
\end{thm}

In \cite{MPP1} we gave two $q$-analogues of Naruse's formula in terms
of SSYT of skew shape, as an identity for $s_{\lambda/\mu}(1,q,q^2,\ldots)$, and for
RPP of skew shape.

\section{Multivariate path identity}\label{sec:multi}


\subsection{Multivariate sums of excited diagrams}
For the skew shape $\lambda/\mu \subseteq d \times (n-d)$ we define $F_{\lambda/\mu}({\bf x}
\,\vert\, {\bf y})$ and $G_{\lambda/\mu}({\bf x} \,\vert\, {\bf y})$ to be the multivariate sums of excited diagrams

\[
G_{\lambda/\mu}({\bf x} \,\vert\, {\bf y}) \, := \,\sum_{D \in \ED(\lambda/\mu)}
  \prod_{(i,j) \in D} (x_i-y_j), \qquad 
F_{\lambda/\mu}({\bf x} \,\vert\, {\bf y})  \, := \, \sum_{D \in \ED(\lambda/\mu)}
\prod_{(i,j)\in [\lambda]\setminus D} \frac{1}{x_i-y_j}\,.
\]

By Proposition~\ref{prop:excited2nips}, the sum \ts $F_{\lambda/\mu}({\bf x}\,\vert\,
{\bf y})$ \ts can be written as a multivariate sum of non-intersecting
  paths.



\begin{cor} \label{cor:FisNIP}  In the notation above, we have:
\[
F_{\lambda/\mu}({\bf x}\,\vert\, {\bf y}) \, = \. \sum_{\Gamma \in
  \NIP(\lambda/\mu)} \prod_{(i,j) \in \Gamma} \frac{1}{x_i-y_j}\,.
\]
\end{cor}


Note that by evaluating $(-1)^{|\lambda/\mu|}F_{\lambda/\mu}({\bf x} \,\vert\, {\bf y})$ at
$x_i = \lambda_i-i+1$ and $y_j=-\lambda'_j+j$ and multiplying by
$|\lambda/\mu|!$ we obtain the right hand side of \eqref{eq:Naruse}.
\begin{equation} \label{eq:relFandf_lam/mu}
\left. (-1)^{|\lambda/\mu|}F_{\lambda/\mu}({\bf x} \,\vert\, {\bf y}) \right|_{\substack{x_i=
    \lambda_i-i+1\\y_j=-\lambda'_j+j}} \, \.  = \, \frac{f^{\lambda/\mu}}{|\lambda/\mu|!}\,.
\end{equation}


Let ${\bf   z}^{\<\lambda\>}$ be the tuple of length $n$ of $x$'s and $y$'s by
reading the horizontal and vertical steps of~$\lambda$ from $(d,1)$ to
$(1,n-d)$:  i.e. $z_{\lambda_i+d-i+1} = x_i$ and
$z_{\lambda'_j+n-d-j+1}=y_j$. For example, for $d=4$, $n=9$
and $\lambda = (5533)$, we have \ts
${\bf z}^{\<\lambda\>} = (y_1,y_2,y_3,x_4,x_3,y_4,y_5,x_2,x_1)$ (see Figure~\ref{fig:path_symmetries}~(i))


\medskip

Combining results of Ikeda--Naruse \cite{IN}, Knutson--Tao
\cite{KT}, and  Lakshmibai--Raghavan--Sankaran, one obtains
the following formula for an evaluation of factorial Schur functions.

\begin{lem}[Theorem~2 in~\cite{IN}] \label{lem:key_border_strips}
For every skew shape \ts $\lambda/\mu \subseteq d\times (n-d)$, we have:
\begin{equation} \label{eq:G2factschur}
G_{\lambda/\mu}({\bf x} \,\vert\, {\bf y}) \, = \,
 {s_{\mu}^{(d)}({\bf x} \,\vert\, {\bf
    z}^{\<\lambda\>})}\..
\end{equation}
\end{lem}


\begin{cor}  We have $F_{\lambda/\mu}({\bf x} \,\vert\, {\bf y}) \, = \, {s_{\mu}^{(d)}({\bf x} \,\vert\, {\bf
    z}^{\<\lambda\>})}/{s_{\lambda}^{(d)}({\bf x} \,\vert\, {\bf
    z}^{\<\lambda\>})}\..$
\end{cor}


\subsection{Symmetries}\label{ss:multi-sym}
%
The factorial Schur function $s^{(d)}_{\mu}({\bf x} \,\vert\, {\bf y})$ is
symmetric in~$\bx$. By Lemma~\ref{lem:key_border_strips}, the
multivariate sum $G_{\lambda/\mu}({\bf x}\,\vert\, {\bf y})$ is an
evaluation of a certain factorial Schur function, which in general
is not symmetric in $\bx$.

\begin{ex}
The shape $\lambda/\mu = 332/21$ from Figure~\ref{ex:excited-diagrams}~(i) has five excited diagrams. One can check that the multivariate polynomial
\begin{multline*}
G_{332/21}(x_1,x_2,x_3\,\vert\, y_1,y_2,y_3) \, =  \, (x_1-y_1)(x_1-y_2)(x_2-y_1) \. + \.
(x_1-y_1)(x_1-y_2)(x_3-y_2) \\ \qquad \quad + \.(x_1-y_1)(x_2-y_3)(x_2-y_1) \. + \.
(x_1-y_1)(x_2-y_3)(x_3-y_2) \. + \.(x_2-y_2)(x_2-y_3)(x_3-y_2)\ts,
\end{multline*}
is not symmetric in  $\bx = (x_1,x_2,x_3)$.
\end{ex}

Now, below we present two cases when the sum
$G_{\lambda/\mu}({\bf x} \,\vert\, {\bf y})$ is in fact symmetric in~$\bx$.
The first case is when~$\mu$ is a rectangle contained in~$\lambda$.

\begin{prop} \label{prop:sym2}
Let $\mu = p^k$ be a rectangle, $p\geq k$, and let $\lambda$ be arbitrary
partition containing~$\mu$.  Denote \ts $\ell:=\max\{i: \lambda_i-i \geq p-k\}$.
Then:
$$
G_{\lambda/p^k}({\bf x} \,\vert\, {\bf y}) \, = \,
s_{p^k}^{(\ell)}(x_1,\ldots,x_\ell \,\vert\, y_1,\ldots,y_{p+\ell-k})\ts.
$$
In particular, the polynomial \ts $G_{\lambda/p^k}({\bf x} \,\vert\, {\bf y})$\ts
is symmetric in $(x_1,\ldots,x_{\ell})$.
\end{prop}
\begin{proof}[Proof sketch]
First, observe that \ts $\ED(\la/p^k) = \ED( (p+\ell-k)^\ell/p^k)$ \ts since the
movement of the excited boxes is limited by the position of the corner
box of~$p^k$, which moves along the diagonal $j-i = p-k$ up to the
boundary of $\la$, at position $(\ell, p+\ell-k)$. Then:
\begin{align*}
G_{\lambda/p^k}({\bf x} \,\vert\, {\bf y})  & \, = \, \sum_{D \in
  \ED((p+\ell-k)^\ell/p^k)} \prod_{(i,j)\in D} (x_i-y_j) \, = \, G_{(p+\ell-k)^\ell/p^k}({\bf x} \,\vert\, {\bf y}) \\
  & \, = \,
  s_{p^k}^{\ell}\bigl(x_1,\ldots,x_{\ell} \,\vert\, y_1,\ldots,y_{p+\ell-k},
x_{\ell},\ldots,x_1\bigr)\..
\end{align*}
Let us now invoke the original combinatorial
formula for the factorial Schur functions,
equation~\eqref{eq:fschur-explicit}, with $a_j = y_j$ for $j\leq
p+\ell-k$ and $a_{p+\ell-k+j} = x_{\ell+1-j}$ otherwise. Note also that
when $T$ is an SSYT of shape $p^k$ and entries at most~$\ell$, by the
strictness of columns we have $T(i,j) \leq \ell-(k-i)$ for all
entries in row $i$. We conclude that $T(i,j) + c(i,j) \leq \ell-k +p$.
Therefore, \ts $a_{T(u)+c(u)}=y_{T(u)+c(u)}$, where only the first
$p+\ell-k$ parameters $a_i$ are involved in the formula. Then:
\begin{align*}
s_\mu^{(\ell)}(x_\ell,\ldots,x_1\,\vert\, a_1,\ldots,a_{p+\ell-k},a_{p+\ell-k+1},\ldots)
& \, = \, s_\mu^{(\ell)}(x_\ell,\ldots,x_1\,\vert\, a_1,\ldots,a_{p+\ell-k}),
\end{align*}
 since now the parameters of the factorial Schur are independent
 of the variables ${\bf x}$ and the function is also symmetric in ${\bf x}$.
\end{proof}




\begin{ex}
For  $\lambda/\mu = 333/22$, the multivariate sum $G_{333/22}(x_1,x_2,x_3 \,\vert\,
y_1,y_2,y_3)$ of the six excited diagrams in
Figure~\ref{ex:excited-diagrams}~(ii) is
symmetric in $x_1,x_2,x_3$.
\end{ex}

\subsection{Multivariate path identity} \label{ss:multi-id}
%
We give an identity for a multivariate sum over non-intersecting
paths as applications of Propositions~\ref{prop:sym2}.






\begin{thm}\label{thm:thickstrip}
We have the following identity for multivariate rational functions:
\begin{equation}
\label{eq:Naruse-1-path}
\sum_{\substack{\Gamma=(\ga_1,\ldots,\ga_c) \\ \ga_p:(a+p,1)\to
  (p,b+c)}} \prod_{(i,j) \in \Gamma} \frac{1}{x_i-y_j} \ \, =
   \sum_{\substack{\Theta=(\theta_1,\ldots,\theta_c)\\ \theta_p:(p,1)
  \to (a+p,b+c)}} \prod_{(i,j) \in \Theta} \frac{1}{x_i-y_j}\,,
 \end{equation}
where the sums are over non-intersecting lattice paths as above.
Note that the left hand side is equal to \ts $F_{(b+c)^{a+c}/b^a}({\bf x} \,\vert\, {\bf y})$ \ts
defined above.
\end{thm}


\begin{wrapfigure}{r}{0.15\textwidth}
  \begin{center}

  \end{center}
\end{wrapfigure}




\begin{figure}
\begin{center}
\includegraphics{pic_path_lemma_fpsac}
\caption{(i) Example of how to read off $z^{\<\lambda\>}$, (ii) paths
  in proof of Theorem~\ref{thm:thickstrip}, and (iii) the shifted hook
shape $\LAM(a,c,d,m)$ appearing in Conjecture~\ref{conj:shiftedshapes}.}
\label{fig:path_symmetries}
\end{center}
\end{figure}


In the next section we use this identity
to obtain product formulas for $f^{\lambda/\mu}$ for certain
families of shapes $\lambda/\mu$.  In the case $c=1$,
we evaluate \eqref{eq:Naruse-1-path} at $x_i =i$ and $y_j = -j+1$
obtain the following corollary.

\begin{cor}[\cite{MPP1}] \label{cor:Naruse-1-path}  We have:
\begin{equation}
\sum_{\ga: (a,1) \to (1,b)} \prod_{(i,j) \in \ga} \frac{1}{i+j-1}  \  =
\sum_{\ga:(1,1) \to (a,b)}\prod_{(i,j) \in \ga} \frac{1}{i+j-1}\,.
\end{equation}
\end{cor}


Equation \eqref{eq:Naruse-1-path} is a
special case of \eqref{eq:Naruse} for the skew shape $(b+1)^{a+1}/b^a$
\cite[Section 3.1]{MPP1}. This equation is also a special case of {\em
Racah formulas} in \cite[Section 10]{BGR} (see \cite[Section 9]{MPP3}).

\begin{proof}[Proof of Theorem~\ref{thm:thickstrip}]
By Proposition~\ref{prop:sym2} for the shape
$(b+c)^{a+c}/b^a$, we have:
\[
G_{(b+c)^{a+c}/b^a}({\bf x} \,\vert\, {\bf y}) \, = \, s_{b^a}(x_1,\ldots,x_{a+c}
\, \,\vert\, \, y_1,\ldots,y_{b+c})\..
\]
Divide the left hand side by $\prod_{(i,j) \in (b+c)^{a+c}} (x_i-y_j)$ to obtain
$F_{(b+c)^{a+c}/b^a}({\bf x} \,\vert\, {\bf y})$, the
multivariate sum over excited diagrams. By
Corollary~\ref{cor:FisNIP}, this is also a multivariate sum over
tuples of non-intersecting paths in $\NIP( (b+c)^{a+c}/b^a)$~:
%
\begin{equation} \label{eq:sumpaths2factschur}
\sum_{\substack{\Gamma=(\ga_1,\ldots,\ga_c) \\ \ga_p:(a+p,1)\to
  (p,b+c)}} \prod_{(i,j) \in \Gamma} \frac{1}{x_i-y_j} \, =\,  s_{b^a}(x_1,\ldots,x_{a+c}
\, \,\vert\, \, y_1,\ldots,y_{b+c}) \prod_{(i,j) \in (b+c)^{a+c}} \frac{1}{x_i-y_j}\,.
\end{equation}
%
Finally, the symmetry in $x_1,\ldots,x_{a+c}$ of the right hand side above
implies that we can flip these variables and
consequently the paths $\ga'_p$ to paths $\theta_p:(p,1)
  \to (a+p,b+c)$ (see Figure~\ref{fig:path_symmetries}~(ii)),
  and obtain the needed
  expression.
\end{proof}


% In the rest of the section we assume that $\lambda/\mu$ is a slim
% shape (see Section~\ref{ss:not-yd}).


\section{Skew shapes with product formulas} \label{sec:skewprod}

In this section we use Theorem~\ref{thm:thickstrip} to obtain product formulas for a
family of skew shapes.

\subsection{Six-parameter family of skew shapes}

For all \ts $a,b,c,d,e,m \in \nn$, let \ts $\LA(a,b,c,d,e,m)$ \ts denote
the skew shape $\lambda/b^a$, where $\lambda$ is 
\begin{equation} \label{eq:deflam}
\lambda \. := \. (b+c)^{a+c} \. + \. \bigl(\nu \cup \theta'\bigr)\.,
\end{equation}
and where $\ts \nu  =(d+(a+c-1)m, d+(a+c-2)m,\ldots,d)$, \ts
$\theta=(e+(b+c-1)m,e+(b+c-2)m,\ldots,e)$; see Figure~\ref{fig:intro_shapes}~(iv). \ts
%
This shape satisfies two key properties:
\begin{align}
\lambda_{a+c+1} &\leq b+c \tag{P1} \label{prop1:lam}\ts, \\
 \lambda_i+\lambda'_j&=\lambda_{r}+\lambda'_s, \quad \text{ if }
                       i+j=r+s \text{ and }   (i,j) \in
(b+c)^{a+c}\ts.   \tag{P2} \label{prop2:lam}
\end{align}
The second property is equivalent to saying that the antidiagonals in
$((b+c)^{a+c})$ inside $\lambda$ have the same
hook-lengths, or that $\ts \lambda_i - \lambda_{i+1} =
\lambda_{i+1}-\lambda_{i+2}$, i.e.\ the parts of $\lambda$ are given by
an arithmetic progression.
% We call these shapes $(a,b,c,d,e,m)$-{\bf  skew shapes}.
Here are two extreme special cases:
$$\LA(a,b,c,0,0,1) \. = \. \delta_{a+b+2c}/b^a\,, \qquad \LA(a,b,c,d,e,0) \. = \. (b+c+d)^{a+c}(b+c)^{e}/b^a\..
$$


Next, we give a product formula for $f^{\pi}$ where
  $\pi=\LA(a,b,c,d,e,m)$ in terms of falling superfactorials
  $\Psi^{(m)}(n) \, := \, \prod_{i=1}^{n-1}\. \prod_{j=1}^i \,
  (jm+j-1)_m$ where $(k)_r = k(k-1)\cdots (k-r+1)$.

\begin{thm} \label{thm:skewprod}
Let \ts $\pi=\LA(a,b,c,d,e,m)$ \ts be as above. Then \ts
$f^{\pi}$ \ts is given by the following product:
%
\begin{multline*} 
f^{\pi} \, = \, \. n! \. \cdot \.
\frac{\Phi(a+b+c)\.\Phi(a)\.\Phi(b)\.\Phi(c)}{\Phi(a+b)\.\Phi(b+c)\.\Phi(a+c)\.\Psi^{(m)}(a+c)\.\Psi^{(m)}(b+c)}
\,\. \times 
\\
\times \prod_{i=0}^{a+c-1} \frac{(i(m+1))!}{(d+i(m+1))!} \, \. \prod_{i=0}^{b+c-1} \frac{(i(m+1))!}{(e+i(m+1))!}
\, \,
\frac{\prod_{i=0}^{b-1} \prod_{j=0}^{a-1} (1+d+e+(c+i+j)(m+1))}{\prod_{i=0}^{b+c-1}
  \prod_{j=0}^{a+c-1} (1+d+e+(i+j)(m+1))}\,.
\end{multline*}
\end{thm}


\smallskip

\begin{proof}[Proof of Corollaries~\ref{cor:abcde-shape},
  \ref{cor:abc-shape} and \ref{cor:abcde1-shape}]
Use~Theorem~\ref{thm:skewprod} for shapes $\LA(a,b,c,e,d,0)$, $\LA(a,b,c,0,0,1)$, and $\LA(a,b,c,d,e,1)$,
respectively.
\end{proof}





The rest of the section is devoted to the proof of
Theorem~\ref{thm:skewprod}. 



\subsection{Proof of the product formulas for skew SYT}

\begin{proof}[Proof of Theorem~\ref{thm:skewprod}]

The starting point is showing that the  skew shape
$\lambda/b^a = \LA(a,b,c,d,e,m)$ and the thick reverse hook
\ts $(b+c)^{a+c}/b^{a}=\LA(a,b,c,0,0,0)$ \ts have the
same excited diagrams. To simplify the notation, let $R=(b+c)^{a+c}$
be the rectangle \ts $\bigl[(a+c)\times (b+c)\bigr]$.

\begin{lem} \label{lem:same_ED}
The skew shapes $\LA(a,b,c,d,e,m)$ and $\LA(a,b,c,0,0,0) =
R/b^a$ have the same excited diagrams.
\end{lem}

\begin{proof}
From the description of excited diagrams:
 by \eqref{prop1:lam}, the cell $(b,a)$ of $[\mu]$ cannot go past the cell $(b+c,a+c)$
so the rest of $[\mu]$ is confined in the rectangle $(b+c)^{a+c}$. 
\end{proof}

By \eqref{eq:Naruse} and Lemma~\ref{lem:same_ED} we have:
\begin{equation} \label{eq2:pfskewprod}
\frac{f^{\lambda/b^a}}{n!}\, = \, \left[\prod_{u\in [\lambda] \setminus
    R} \, \frac{1}{h_{\lambda}(i,j)}\right] \, \sum_{D \in
  \ED(R/b^a)}  \prod_{(i,j) \in R\setminus D}\frac{1}{h_{\lambda}(i,j)}\,.
\end{equation}

The sum over excited diagrams of $R/b^a$
  with hook-lengths in $\lambda$ on the right hand side above evaluates to a
  product.

\begin{lem}  \label{lem:sumED2prod}
For $\lambda$ and $R$ as above we have:
\begin{equation}
\sum_{D \in
  \ED(R/b^a)} \prod_{(i,j) \in R\setminus D} \frac{1}{h_{\lambda}(i,j)} \,=\, \frac{\Phi(a+b+c)\.\Phi(a)\.
  \Phi(b)\.\Phi(c)}{\Phi(a+b)\.\Phi(b+c)\.\Phi(a+c)} \, \prod_{(i,j) \in R/0^cb^{a}}\,\frac{1}{h_{\lambda}(i,j)}\,.
\end{equation}
\end{lem}

\begin{proof}
We write the sum of excited diagrams as an evaluation of \ts $F_{(b+c)^{a+c}/b^a}({\bf x} \,\vert\, {\bf y})$.
\begin{equation}\label{eq3:pfskewprod}
\sum_{D \in
  \ED(R/b^a)} \prod_{(i,j) \in R\setminus D} \frac{1}{h_{\lambda}(i,j)} \,=\,
\left. (-1)^m F_{(b+c)^{a+c}/b^{a}}({\bf x} \,\vert\, {\bf y}) \right|_{\substack{x_i = \lambda_i  -i+1 \\y_j =j-\lambda'_j}}
\end{equation}
where $m=(b+c)(a+c)-ba$. Using Theorem~\ref{thm:thickstrip} to obtain
the symmetry of the series $F_{(b+c)^{a+c}/b^a}({\bf x} \,\vert\, {\bf y})$ in~$\bx$~:
\begin{equation} \label{eq4:pfskewprod}
\left. (-1)^mF_{(b+c)^{a+c}/b^{a}}({\bf x} \,\vert\, {\bf y}) \right|_{\substack{x_i= \lambda_i -i+1 \\y_j=j-\lambda'_j}} \, =\,
\sum_{\Theta} \prod_{(i,j) \in \Theta} \frac{1}{h_{\lambda}(i,j)}\,,
\end{equation}
where the sum is over tuples $\ts \Theta:=(\theta_1,\ldots,\theta_c)$ \ts of
nonintersecting paths inside $(b+c)^{a+c}$ with endpoints \ts
$\theta_p:(p,1) \to (a+p,b+c)$. Note that each tuple $\Theta$ has the same
number of cells in each diagonal $i+j=k$. Also, by property \eqref{prop2:lam}
of~$\lambda$, the sum \ts $(\lambda_i+\lambda'_j)$ \ts is constant
when \ts $(i+j)$ \ts is constant. Thus each tuple $\Theta$
will have the same contribution to the sum on the right hand side of
\eqref{eq4:pfskewprod}, namely
\[
\prod_{(i,j) \in \Theta} \. \frac{1}{h_{\lambda}(i,j)} \,=\,
\prod_{(i,j) \in R/0^cb^{a}} \frac{1}{h_{\lambda}(i,j)} \,.
\]
Lastly, the number of tuples $\Theta$ in \eqref{eq4:pfskewprod} equals the number of
excited diagrams $(b+c)^{a+c}/b^{a}$, given by \eqref{eq:macmahon}.
\end{proof}


By Lemma~\ref{lem:sumED2prod}, \eqref{eq2:pfskewprod} becomes the following product formula for
$f^{\lambda/b^a}$~:
\begin{equation} \label{eq6:pfskewprod}
\frac{f^{\lambda/b^a}}{n!} \, = \, \frac{\Phi(a+b+c)\.\Phi(a)\.\Phi(b)\.
\Phi(c)}{\Phi(a+b)\.\Phi(b+c)\.\Phi(a+c)} \,
\prod_{(i,j) \in \lambda/0^cb^{a}} \, \frac{1}{h_{\lambda}(i,j)} \,.
\end{equation}
Finally, we carefully rewrite this product in terms of $\Psi(\cdot)$ and
$\Psi^{(m)}(\cdot)$.
\end{proof}



\section{Final remark and a conjecture}

In the full version \cite{MPP3} of this paper we give
$q$-analogues of our product formulas.

It is natural to study shifted analogues of our product formulas. For nonnegative integers $a\le c,d$ and~$m$, let $\lambda/\mu=\LAM(a,c,d,m)$
be the following shifted skew partition,
\[
\lambda \. = \. (c+d, c+d-1,\ldots,c+d-a+1) \. + \. \nu\ts,
\]
where $\ts \nu = (d+(a+c-1)m,d+(a+c-2)m,\ldots,d)$ \ts and $\ts \mu =
\delta_{a+1}$. See Figure~\ref{fig:path_symmetries}~(iii).

\begin{conj} \label{conj:shiftedshapes}
In the notation above, the number of SYT of shape $\pi =
\LAM(a,c,d,m)$ equals
$$g^{\pi}  \, = \, \frac{n!}{2^{a}} \, \cdot \,
\frac{\Phi(2a+c) \. \Phi(a)}{\Phi(2a)\. \Phi(a+c)} \, \cdot \,
\frac{\Lam(2a) \. \Lam(c)}{\Lam(2a+c)} \, \cdot \,
\prod_{u \in \lambda \setminus (\delta_{a+c+1}/c^a\delta_{c+1})} \, \frac{1}{h_B(u)} \,.
$$
\end{conj}

A special case \ts $\pi = \LAM(a,c,d,0)$ \ts is the (conjugated)
\emph{truncated rectangle shape}, and was established by the third author
using a different technique~\cite{Pan}. 
%We would be ever sensible of
%the warmest gratitude towards the persons who had been the means of
%uniting these product formulas.




\acknowledgements{We are grateful to
Dan Betea, Sara Billey, Zach Hamaker, Valentin F\'eray, 
Luis Serrano, Richard Stanley, Nathan Williams,
Damir Yeliussizov and Alex Yong for useful comments and help with the references.
We are very thankful to Christian Krattenthaler for
pointing out to us paper~\cite{KS}, and to Jang Soo Kim for pointing us out that
Corollary~\ref{cor:abcde-shape} appeared in~\cite{KO}.}

%% if you use biblatex then this generates the bibliography
%% if you use some other method then remove this and do it your own way



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\subsection{SSYT}

In \cite{MPP3}, we also give a product formula for the generating function of SSYT of
these shapes. As in the proof above, we obtain explicit formulas for the skew shapes
$\LA(a,b,c,d,e,0)$, $\LA(a,b,c,0,0,1)$, and $\LA(a,b,c,d,e,1)$. These
formulas can be found in the full version of \cite{MPP3} and are
written in terms of $q$-versions of the functions in \eqref{eq:defsuperfact}. Up to the power of $q$, these
cases are obtained by ``$q$-ifying'' their counterpart formulas for
$f^{\lambda/\mu}$. The $q$-analogue of the case $\pi=\LA(a,b,c,0,0,1)$
appeared in work of Krattenthaler--Schlosser~\cite{KS}.

\subsection{Asymptotics}


In \cite{MPP4} we use this second symmetry identity to give new lower bounds on $f^{\lambda/\mu}$
for several other families of slim shapes $\lambda/\mu$.

