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\title{Set-valued tableaux for Macdonald polynomials}



%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author{Zajj Daugherty\thanks{\href{mailto:zdaugherty@gmail.com}{zdaugherty@reed.edu}}\addressmark{1}
\and 
Arun Ram\thanks{\href{mailto:aram@unimelb.edu.au}{aram@unimelb.edu.au}}\addressmark{2}}
%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Department of Mathematics and Statistics, Reed College,  
Portland, OR 97202\\ \addressmark{2}Department of Mathematics and Statistics, University of Melbourne, Parkville VIC 3010 AU}

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


%% 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{
Set-valued tableaux formulas play an important role in
Schubert calculus. Using the box greedy reduced word for the construction of the Macdonald polynomials,
we convert the alcove walk formula to a set-valued tableaux formula for Macdonald polynomials. Our hope is that providing such formulas will help
to strengthen the analogies and possible connections between the calculus
of Macdonald polynomials and Schubert calculus.
}

%% 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{Macdonald polynomials, tableaux formulas, Schubert calculus}



%% 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; i.e., [xx, yy] in the text

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\begin{document}

\maketitle




\section{Introduction}

The goal of this paper is to give set-valued tableaux formulas for Macdonald polynomials.
As evidenced by the work of \cite{Bu02}, \cite{KMY09}, \cite{MS19}, \cite{SP19}, \cite{Wei20}, \cite{Yu21} and others, 
set-valued tableaux formulas play an important role in
Schubert calculus.   Although there are many hints from the study of affine Springer fibers \cite{Me17} and
seminfinite flag varieties \cite{KNS18} the concrete connection between Schubert calculus and
Macdonald polynomials for general type and general $q$ and $t$ is still elusive.  
There are many flavors of set-valued tableaux in the Schubert calculus
literature and the relationships between them, including the relationship to the set-valued tableaux that we use, 
remain under-explored.  Our hope is to help strengthen the analogies and possible connections between the calculus
of Macdonald polynomials and Schubert calculus.  See \cite{DR22} for a version of this paper
with additional details and examples.

Our formulas for Macdonald polynomials 
are derived by making a bijection between set-valued tableaux
and alcove walks 
and converting the alcove walks formula from \cite[Theorem 2.2]{RY08} to a set-valued tableaux formula.
We follow the framework of \cite[\S1 and \S2]{GR21}, which gives an exposition of the alcove walks formula
in the $GL_n$ context and analyzes the favourite reduced word for the $n$-periodic permutation $u_\mu$
that is used to construct the relative Macdonald polynomial $E^z_\mu$.
Then a set-valued tableau formula for
the symmetric Macdonald polynomials $P_\lambda$, with $\lambda$ a partition, follows from 
$$P_\lambda = (const)\sum_{z\in S_n\lambda} E^z_\lambda\ ;$$
see \cite[(1.4),(1.5) and (1.10)]{GR21b}.

The set-valued tableaux formulas given in \cref{pospowers} (respectively, \cref{negpowers}) are specializable
at $q=0$ and $t=0$ (respectively, $q=\infty$ and $t=\infty$).
The specializations of $E^z_\mu$ at $t=0$ and $t=\infty$ have interpretations in terms of crystals
for level 1 and level 0 affine Demazure characters (see, for example, \cite[Theorem C and Corollary D]{Kt16}
or \cite[(2.12), Theorem 1.1 and \S2.4]{MRY19}).  
It would be interesting to write the root operators for these crystals explicitly on the set-valued tableaux
(it is likely that the crystals of \cite[\S4]{Yu21} cover some cases of these crystals).



\section{Set-valued tableaux for Macdonald polynomials}

We begin with combinatorial definitions necessary for stating the set-valued tableaux formula
for Macdonald polynomials.  
To aid in processing these definitions, we encourage the reader to follow
\cref{runningexample} in tandem.

Fix $n\in \ZZ_{>0}$ and let $S_n$ denote the symmetric group on $n$ letters.
Identify $\mu=(\mu_1,\ldots, \mu_n)\in \ZZ_{\ge 0}^n$
with \emph{the set of boxes in $\mu$},
$$\mu = \{ (r,c)\ |\ \hbox{$r\in \{1, \ldots, n\}$ and $c\in \{1, \ldots, \mu_r\}$} \}.$$

\subsection{Definition of set-valued tableaux}\label{eq:u(rc)}

For $(r,c)\in \mu$ define
\begin{equation*}
u_\mu(r,c) = \#\{ r'\in \{1, \ldots, r-1\}\ |\ \mu_{r'}<c\le \mu_r\} + \#\{r'\in \{r+1, \ldots, n\}\ |\ \mu_{r'}<c-1<\mu_i\}.
\end{equation*}
The values $u_\mu(r,c)$ play a similar role to the flagging in the use of set-valued tableaux to give
formulas for Grothendieck polynomials (see \cite[\S2.6 and Theorem 7.1]{Wei20}).
A \emph{set-valued tableau $T$ of shape $\mu$} 
is a choice of subset $T(r,c)\subseteq \{1,\ldots, u_\mu(r,c)\}$ 
for each box $(r,c)\in \mu$.  More formally, a set-valued tableau $T$ of shape $\mu$
is a function 
$$T\colon \mu \to \{ \hbox{subsets of $\{1, \ldots, n\}$}\}
\qquad\hbox{such that}\quad
T(r,c) \subseteq \{1,\ldots, u_\mu(r,c)\}.$$
Let $\vert T\vert$ denote the total number of entries in $T$.


\subsection{Definition of the shift and height statistics} \label{shhtdefn}


The minimal length permutation $v_\mu\in S_n$ such that $v_\mu \mu$ is weakly increasing
is given by
$$v_\mu(r) = 1+\#\{r'\in \{1, \ldots, r-1\}\ |\ \mu_{r'}\le \mu_r\}
+\#\{r'\in \{r+1, \ldots, n\}\ |\ \mu_{r'}<\mu_r\},$$
for $r\in \{1, \ldots, n\}$.
For a box $(r,c)\in \mu$ and $i\in \{1, \ldots, u_\mu(r,c)\}$ define
\begin{equation*}
\mathrm{sh}(r,c) = \mu_r-c+1
\qquad\hbox{and}\qquad
\mathrm{ht}(i,r,c) 
= v_\mu(r) - i.
\end{equation*}


\subsection{Definition of $z^T_{(r,c)}$ and $x^T$} \label{zTrcdefn}


Using one-line notation, let  $\gamma_n$ be the $n$-cycle in the symmetric group $S_n$ given by
$$\gamma_n  
= \begin{pmatrix} 
n &1 &2 &\cdots &n-1\end{pmatrix}.$$
For positive integers $k_1, \ldots, k_\ell$ such that $k_1+\cdots+k_\ell = n$, let 
$$\gamma_{k_1}\times \cdots \times \gamma_{k_\ell}
\qquad\hbox{be the disjoint
product of cycles in}\qquad
S_{k_1}\times \cdots \times S_{k_\ell}\subseteq S_n.
$$

Fix a permutation $z\in S_n$.  
Order the boxes of $\mu$ down columns and then left to right (i.e., in increasing order of the values $r+nc$).
Starting from the permutation $z$,
associate a permutation $z^T_{(r,c)}$ to each box $(r,c)$ of $\mu$ as follows: \hfill\break
If $(r,c)\in \mu$ 
and $T(r,c) =\{m_1, \ldots, m_p\}$ with $1\le m_1<\cdots <m_p\le u_\mu(r,c)$,  
define
\begin{equation*}
z_{(r,c)}^T = z_{(r',c')}^T
(\gamma_{m_1-0}\times  \gamma_{m_2-m_1} \cdots \times \gamma_{m_p-m_{p-1}}
\times \gamma_{u_\mu(r,c)+1-m_p}\times (\gamma_1)^{\times (n-u_\mu(r,c)-1)}) \gamma_n^{-1},
\end{equation*}
where $(r',c')\in \mu$ denotes the box before $(r,c)$ in $\mu$.
In the case that $(r,c)$ is the first box in $\mu$ we let $z^T_{(r',c')} = z$.
Then define 
\begin{equation*}
x^T = \prod_{(r,c)\in \mu} x_{z^T_{(r,c)}(n)},
\end{equation*}
where $z^T_{(r,c)}(n)$ is the value of the permutation $z^T_{(r,c)}$ at $n$.

The permutation $z^T_{\mathrm{init}}=z$ is the \emph{left key} (or \emph{initial direction}) of $T$ and
the permutation $z^T_{\mathrm{fin}}$ corresponding to the last box in $\mu$ 
is the \emph{right key} (or \emph{final direction}) of $T$.
These left and right keys play important roles in Schubert calculus: for example in 
\cite[\S3]{Yu21} and in the statement of Pieri-Chevalley formulas 
(compare \cite[Theorem 1]{FL94} and \cite[Corollary]{PR98}).

\subsection{Definition of the cov and maj statistics.}\label{pnfolds}

Keep the notation $T(r,c) =\{m_1, \ldots, m_p\}$ with $m_1<\ldots<m_p$.
Define
\begin{align*}
T^z_<(r,c) &= \{ m_k \in T(r,c)\ |\ 
 z^T_{(r,c)}(m_{k-1} ) < z^T_{(r,c)}(m_k)\}
\qquad\hbox{and} 
\nonumber \\
T^z_>(r,c) &= \{ m_k\in T(r,c)\ |\ 
z^T_{(r,c)}(m_{k-1} ) > z^T_{(r,c)}(m_k) \},
\end{align*}
where we make the convention that $m_0=n$.
Then define
\begin{align*}
\mathrm{maj}_>^z(T) 
&
= \sum_{(r,c)\in \mu} \mathrm{sh}(r,c)\cdot \vert T^z_>(r,c)\vert, \\
\mathrm{cov}_>^z(T) 
&= \Big( \sum_{(r,c)\in \mu} \sum_{m\in T^z_>(r,c)} \mathrm{ht}(m,r,c) \Big)
+ \hbox{$\frac12$} \big(\ \ell(z^T_{\mathrm{fin}}) - \ell(z^T_{\mathrm{init}}v_\mu^{-1}) - \vert T\vert \ \big), 
\end{align*}
where $\ell(w)$ denotes the length of the permutation $w$ in $S_n$. 


\subsection{Set-valued tableaux formulas for $E^z_\mu$.}

The relative Macdonald polynomial $E^z_\mu = E^z_\mu(x_1, \ldots, x_n;q,t)$
is what is termed the ``permuted basement nonsymmetric Macdonald polynomial'' in \cite{Al16}
(see \cite[(3.7)]{GR21} for further references and a definition in terms of Cherednik-Dunkl operators
and the double affine Hecke algebra action on polynomials).

\begin{thm}  \label{pospowers}
Let $z\in S_n$ and $\mu\in \ZZ_{\ge 0}^n$.  Then the relative Macdonald polynomial 
$E^z_\mu$ is
$$E^z_\mu = \sum_T  q^{\mathrm{maj}_>^z(T)}t^{\mathrm{cov}_>^z(T)}
\Big(\prod_{(r,c)\in\mu} \prod_{m\in T(r,c)} \frac{1-t}{1-q^{\mathrm{sh}(r,c)}t^{\mathrm{ht}(m,r,c)}}\Big)
x^T,
$$
where the sum is over set-valued tableaux $T$ of shape $\mu$.
\end{thm}

Because the powers of $q$ and $t$ in formula in \cref{pospowers} are nonnegative integers, this formula
is well-suited to the specializations $q=0$ and/or $t=0$.  The following theorem provides an alternate
formula which is better for identifying the specializations of $E^z_\mu$ at $q=\infty$ and/or $t=\infty$ (i.e.,
specializations at $q^{-1}=0$ and $t^{-1}=0$).
\begin{align*}
\mathrm{maj}^z_<(T) 
&
= \sum_{(r,c)\in \mu} \mathrm{sh}(r,c)\cdot \vert T^z_<(r,c)\vert, \\
\mathrm{cov}^z_<(T) 
&= \Big( \sum_{(r,c)\in \mu} \sum_{m\in T^z_<(r,c)} \mathrm{ht}(m,r,c) \Big)
- \hbox{$\frac12$} \big(\ \ell(z^T_{\mathrm{fin}}) - \ell(z^T_{\mathrm{init}}v_\mu^{-1}) - \vert T\vert \ \big).
\end{align*}

\begin{thm}  \label{negpowers} 
Let $z\in S_n$ and $\mu\in \ZZ_{\ge 0}^n$.  Then the relative Macdonald polynomial 
$E^z_\mu$ is
$$E^z_\mu = \sum_T  q^{-\mathrm{maj}^z_<(T)}t^{-\mathrm{cov}^z_<(T)}
\Big(\prod_{(r,c)\in\mu} \prod_{m\in T_{(r,c)}} 
\frac{1-t^{-1}}{1-q^{-\mathrm{sh}(r,c)}t^{-\mathrm{ht}(i,r,c)}}\Big)
x^T,
$$
where the sum is over set-valued tableaux $T$ of shape $\mu$.
\end{thm}

The proof of \cref{negpowers} is obtained from \cref{pospowers} by multiplying
numerator and denominator of each coefficient by
$$
\prod_{(r,c)\in \mu}\prod_{i=1}^{u_\mu(r,c)} q^{-\mathrm{sh}(r,c)}t^{-\mathrm{ht}(i,r,c)}.
$$
The proof of \cref{pospowers} is obtained by considering the reduced word for the $n$-periodic
permutation
$$u_\mu = \prod_{(r,c)\in \mu} s_{u_\mu(r,c)-1}\cdots s_2s_1\pi
\qquad \hbox{(product taken in increasing order of boxes),}
$$
studied in \cite[Propositions 2.1 and 2.2]{GR21} (see also \cite[(2.4.3)]{Mac03}) and
using a straightforward bijection between alcove walks of type $u_\mu$ and set-valued tableaux of shape $\mu$
(see \cref{sec:conversion}).
This bijection gives a conversion between the alcove walk formula for in $E^z_\mu$
in \cite[Theorem 1.1(a)]{GR21} (see also \cite[Theorem 2.2]{RY08} for this alcove walk formula in a root system language) 
and the set-valued tableaux formula in \cref{pospowers}.


\begin{example} \label{runningexample}
Let $n=5 $ and $\mu = (0,4,5,1,4)$. Then the box arrangement associated to $\mu$ is as follows, with boxes placed on a labeled grid, filled with values $u_\mu(r,c)$, and marked in the lower corner by their cylindrical numbers $r+nc$.

\begin{equation}\label{eq:Box arrangement example}
\TikZ{[xscale=.75,yscale=-.75]	
	\foreach \x in {1,2,3,4,5}{
		\draw[black!70, very thin] (\x, -.2) to (\x, 5.2);
		\node[black!70,above] at (\x-.5, 0) {\tiny $\x$};}
	\draw[black!70, very thin] (0, -.2) to (0, 5.2);
	\foreach \x in {1,2,...,5}{
		\draw[black!70, thin] (-.2,\x) to (5.2,\x);
		\node[black!70,left] at (0, \x-.5) {\tiny $\x$};}
	\draw[black!70, thin] (-.2,0) to (5.2,0);
	\node[black!70, above] at (3-.5, -.5) {\scriptsize column $c$};
	\node[black!70, left ] at (-.5, 3-.5) {\scriptsize row $r$};
	\setcounter{bx}{1}
	\foreach \Col [count = \x from 1] in {{2,3,4,5}, {2,3,5},{2,3,5}, {2,3,5}, {3}}{
	\foreach \y in \Col {
		\setcounter{bx}{\x}
		\foreach \k in {1,2,3,4}{\addtocounter{bx}{\x}}
		\addtocounter{bx}{\y}
		\node[inner sep = 1pt, above right, dblue] at (\x-1, \y) {\tiny \thebx};
		\draw[dblue] (\x-1+.425, \y) .. controls +(0,-.325) .. +(-.425,-.325);
		}
		}
        \begin{scope}[thick]
        \smTableau{
        {},
        {{~1}, {~1}, {~2}, {~2}},
        {{~1},{~1},{~2},{~2},{~3}},
        {{~1}},
        {{~1},{~2},{~2},{~2}}}
	%missing  edges:
	\draw[] (0,0) to (0,1) to (4,1) (4,2) to (5,2) (1,4) to (4,4);
	\end{scope}
}
\end{equation}
Box by box, the corresponding factors $\frac{(1-t)}{1-q^{\mathrm{sh}(r,c)}t^{\mathrm{ht}(i,r,c)}}$ are
$$
\def\arraystretch{1.2}
\setlength{\arraycolsep}{3pt}
\frac{(1-t)}{1-q^{\mathrm{sh}}t^{\mathrm{ht}}}
=\begin{array}{|T|T|T|T|T|}
\multicolumn{1}{|T}{\TikZ{\draw[white] (0,0) to (0,.5);}}
\\\cline{1-4}
\begin{matrix} \frac{(1-t)}{(1-q^4t^2)}\end{matrix}
&\begin{matrix} \frac{(1-t)}{(1-q^3t^2)}\end{matrix} 
&\begin{matrix} \frac{(1-t)}{(1-q^2t^2)} \\ \frac{(1-t)}{(1-q^2t)} \end{matrix}
&\begin{matrix} \frac{(1-t)}{(1-qt^2)} \\ \frac{(1-t)}{(1-qt)} \end{matrix}&\multicolumn{1}{T}{}
\\\cline{1-5}
\begin{matrix} \frac{(1-t)}{(1-q^5t^4)}\end{matrix} 
&\begin{matrix} \frac{(1-t)}{(1-q^4t^4)}\end{matrix}
&\begin{matrix} \frac{(1-t)}{(1-q^3t^4)} \\ \frac{(1-t)}{(1-q^3t^3)}\end{matrix}
&\begin{matrix} \frac{(1-t)}{(1-q^2t^4)} \\ \frac{(1-t)}{(1-q^2t^3)}\end{matrix}
&\begin{matrix} \frac{(1-t)}{(1-qt^4)} \\ \frac{(1-t)}{(1-qt^3)} \\ \frac{(1-t)}{(1-qt^2)} \end{matrix}
\\\cline{1-5}
\phantom{\Big|}
\begin{matrix} \frac{(1-t)}{(1-qt)}\end{matrix}
\phantom{\Big|}
\\\cline{1-4}
\begin{matrix} \frac{(1-t)}{(1-q^4t^3)}\end{matrix}
&\begin{matrix} \frac{(1-t)}{(1-q^3t^3)} \\ \frac{(1-t)}{(1-q^3t^2)}\end{matrix}
&\begin{matrix} \frac{(1-t)}{(1-q^2t^3)} \\ \frac{(1-t)}{(1-q^2t^2)} \end{matrix}
&\begin{matrix} \frac{(1-t)}{(1-qt^3)} \\ \frac{(1-t)}{(1-qt^2)} \end{matrix}\\\cline{1-4}
\end{array}\ .
$$


\noindent 
One set-valued tableau of shape $\lambda$ is 
\begin{equation}\label{eq:favoriteT}
T =
\ \TikZ{[xscale=.85,yscale=-.55]
     \smTableau{
        {},
        {{\{1\}}, {\emptyset}, {\{1\}}, {\{2\}}},
        {{\{1\}},{\emptyset},{\{1,\! 2\}},{\emptyset},{\{1,\! 3\}}},
        {{\emptyset}},
        {{\emptyset},{\emptyset},{\{1,\! 2\}},{\emptyset}}}
	%missing edges:
	\draw[] (0,0) to (0,1) to (4,1) (4,2) to (5,2) (1,4) to (4,4);
}
=
\ \TikZ{[xscale=.55,yscale=-.55]
     \smTableau{
        {},
        {{1}, {}, {1}, {2}},
        {{1},{},{1\,2},{},{1\,3}},
        {{}},
        {{},{},{1\,2},{}}}
	%missing edges:
	\draw[] (0,0) to (0,1) to (4,1) (4,2) to (5,2) (1,4) to (4,4);
}
\end{equation}
which has size 
$\vert T \vert = 1+1+1+2+2+1+2 = 10$ (we shall often omit the set notation and commas as in the right-most picture above).
The product in \cref{pospowers} corresponding to this set-valued tableau $T$ is
$$\prod_{(r,c)\in \mu} \prod_{m\in T^z(r,c)} \frac{(1-t)}{1-q^{\mathrm{sh}(r,c)}t^{\mathrm{ht}(i,r,c)}}
=
\def\XA{1.5}\def\XB{2.25}\def\XC{5}\def\XD{6.2}\def\XE{8.7}
\def\YA{0}\def\YB{1}\def\YC{2}\def\YD{3}\def\YE{4}\def\YF{5}
\ \TikZ{[xscale=1,yscale=-1]
\node at (.5*\XA, .5*\YB) {$\frac{(1-t)}{(1-q^4t^2)}$};
	\node at (.5*\XA+.5*\XB, .5*\YB) {$1$};
	\node at (.5*\XB+.5*\XC, .5*\YB) {$\frac{(1-t)}{(1-q^2t^2)}$};
	\node at (.5*\XC+.5*\XD, .5*\YB) {$\frac{(1-t)}{(1-qt)}$};
\node at (.5*\XA, .5*\YB+.5*\YC) {$\frac{(1-t)}{(1-q^5t^4)}$};
	\node at (.5*\XA+.5*\XB, .5*\YB+.5*\YC) {$1$};
	\node at (.5*\XB+.5*\XC, .5*\YB+.5*\YC) {$\frac{(1-t)}{(1-q^3t^4)}\frac{(1-t)}{(1-q^3t^3)}$};
	\node at (.5*\XC+.5*\XD, .5*\YB+.5*\YC) {$1$};
	\node at (.5*\XD+.5*\XE, .5*\YB+.5*\YC) {$\frac{(1-t)}{(1-qt^4)}\frac{(1-t)}{(1-qt^2)}$};
\node at (.5*\XA, .5*\YC+.5*\YD) {$1$};
\node at (.5*\XA, .5*\YD+.5*\YE) {$1$};
	\node at (.5*\XA+.5*\XB, .5*\YD+.5*\YE) {$1$};
	\node at (.5*\XB+.5*\XC, .5*\YD+.5*\YE) {$\frac{(1-t)}{(1-q^2t^3)}\frac{(1-t)}{(1-q^2t^2)}$};
	\node at (.5*\XC+.5*\XD, .5*\YD+.5*\YE) {$1$};
\foreach \x/\y in {\XD/\YA, \XE/\YB, \XE/\YC, \XD/\YD, \XD/\YE}
{\draw (0, \y) to (\x, \y);}
\draw (0,-.75) to (0, \YE) (\XA,0) to (\XA, \YE) 
	(\XB,0) to (\XB, \YC) (\XB, \YD) to (\XB, \YE)
	(\XC,0) to (\XC, \YC) (\XC, \YD) to (\XC, \YE)
	(\XD,0) to (\XD, \YC) (\XD, \YD) to (\XD, \YE)
	(\XE, \YB) to (\XE, \YC);
}\ .
$$

Let $z = \id$ be the identity.  The \emph{box-by-box permutation sequence of $T$ with initial direction $z$} 
(written in 1-line notation $w = (w(1)\ \cdots \ w(n))$) is
\begin{equation}\label{eq:zFavT}
z^T =
\ \TikZ{[xscale=1.35,yscale=-.6]
     \smTableau{
        {},
        {{\Mk{2,3,4,5,\bred{1}}{1}}, 	{\Mk{3,2,4,5,\bred{1}}{}}, 	{\Mk{1,4,2,5,\bred{3}}{1}}, 	{\Mk{2,3,1,4,\bred{5}}{2}}},
        {{\Mk{3,4,5,1,\bred{2}}{1}},		{\Mk{3,4,5,1,\bred{2}}{}},	{\Mk{4,2,5,3,\bred{1}}{1,2}},	{\Mk{2,4,3,5,\bred{1}}{}},	{\Mk{5,3,1,4,\bred{2}}{1,3}}
        },
        {{\Mk{3,5,1,2,\bred{4}}{}}},
        {{\Mk{3,1,2,4,\bred{5}}{}},{(3412\bred{5})},{\Mk{2,5,3,1,\bred{4}}{1,2}},{\Mk{2,3,5,1,\bred{4}}{}}}}
	%missing edges:
	\draw[] (0,0) to (0,1) to (4,1) (4,2) to (5,2) (1,4) to (4,4);
}.
\end{equation}
In each box we have underlined the positions specified by the entries of $T$; these positions are the
$m_1, \ldots, m_p$ in \cref{pnfolds}.  The red highlighted entry indicates $z^T_{(r,c)}(n)$, which is used
in the formula for $x^T$ in \cref{zTrcdefn}, and also as $z^T_{(r,c)}(m_0)$ in the definition of
$T^z_<(r,c)$ and $T^z_>(r,c)$ in \cref{pnfolds}.  So we have
\begin{equation*}
x^{T} =
\TikZ{[xscale=.6,yscale=-.6]
     \nTableau{
        {},
        {{x_{\bred{1}}}, {x_{\bred{1}}}, {x_{\bred{3}}}, {x_{\bred{5}}}},
        {{x_{\bred{2}}},{x_{\bred{2}}},{x_{\bred{1}}},{x_{\bred{1}}},{x_{\bred{2}}}},
        {{x_{\bred{4}}}},
        {{x_{\bred{5}}},{x_{\bred{5}}},{x_{\bred{4}}},{x_{\bred{4}}}}}
	%missing edges:
	\draw[] (0,0) to (0,1) to (4,1) (4,2) to (5,2) (1,4) to (4,4);
}
= x_1^4x_2^3x_3x_4^3x_5^3\ ;
\end{equation*}
and, following the definition in \cref{pnfolds} and
splitting $T$ into \emph{ascents} and \emph{descents} in the sequence of underlined numbers (preceded by the red number $z_{(r,c)}^T(n)$): 
\begin{equation*}
T^z_< = 
\ \TikZ{[xscale=.5,yscale=-.5]
     \nTableau{
        {},
        {{1}, {}, {}, {}},
        {{1},{},{},{},{1}},
        {{}},
        {{},{},{2},{}}}
	%missing edges:
	\draw[] (0,0) to (0,1) to (4,1) (4,2) to (5,2) (1,4) to (4,4);
}
\qquad\text{and}\qquad
T^z_> = \ \TikZ{[xscale=.5,yscale=-.5]
     \nTableau{
        {},
        {{}, {}, {1}, {2}},
        {{},{},{2},{},{3}},
        {{}},
        {{},{},{1},{}}}
	%missing edges:
	\draw[] (0,0) to (0,1) to (4,1) (4,2) to (5,2) (1,4) to (4,4);
}\ .
\end{equation*}
The initial and  final directions of $T$ are
\begin{equation*}
z^T_{\mathrm{init}} = \id = (12345)
\qquad\hbox{and}\qquad
z^T_{\mathrm{fin}} = z^T_{(3,5)} = (53142),
\end{equation*}
where $z_{(3,5)}$ indicates the permutation in box $(3,5)$ of $z^T$.  
Since $v_\mu = (13524)$ then
$$\ell(z^T_{\mathrm{fin}}) = 4+2+0+1 = 7
\quad \text{ and } \quad 
\ell(z^T_{\mathrm{init}}v^{-1}_\mu) = 3,$$
so that 
$$\hbox{$\frac12$}(\ell(z^T_{\mathrm{fin}})-\ell(z^T_{\mathrm{init}}v^{-1}_\mu)-\vert T\vert )
=\hbox{$\frac12$}(7-3-10) = -3.$$
Then
\begin{equation*}
q^{\mathrm{maj}^z_>(T)}t^{\mathrm{cov}^z_>(T)}
= t^{\frac12(7-3-10)}\cdot 
\def\arraystretch{1.1}
\begin{array}{|c|c|c|c|c|}
\multicolumn{1}{|c}{~}
\\\cline{1-4}
1 &1 &q^2t^2 &qt \\\hline
1 &1 &q^3t^3 &1 &qt^2 \\\hline
1 \\\cline{1-4}
1 &1 &q^2t^3 &1\\\cline{1-4}
\end{array}
=t^{-3} q^9 t^{11}
= q^9 t^8.
\end{equation*}
\end{example}



\section{Proof of \cref{pospowers}}

In this section we describe the conversion from alcove walks to set-valued tableaux.
We follow the framework of \cite[\S1]{GR21}, which gives an exposition of the alcove walks formula
in the $GL_n$ context and analyzes the favourite reduced for the $n$-periodic permutation $u_\mu$
that is used to construct the relative Macdonald polynomial $E^z_\mu$.

\subsection{Inversions and the box-greedy reduced word}

An \emph{$n$-periodic permutation} is a bijection $w\colon \ZZ\to \ZZ$ such that
$w(i+n) = w(i)+n$.  Since an $n$-periodic permutation $w$ is determined by the values
$w(1), \ldots, w(n)$,
a permutation $w\in S_n$ extends uniquely to an $n$-periodic permutation.
For $i\in \{1, \ldots, n-1\}$ let $s_i$ be the transposition in $S_n$ switching $i$ and $i+1$.
Define $\pi\colon \ZZ\to \ZZ$ by $\pi(i) = i+1$.

An \emph{inversion} of an $n$-periodic permutation $w$ is an element of the set
$$\mathrm{Inv}(w) = \{ (i,k)\ |\ \hbox{$i\in \{1, \ldots, n\}$, $k\in \ZZ$, $i<k$ and $w(i)>w(k)$}\}.$$
If $i,j\in \{1, \ldots, n\}$ with $i<j$ and $\ell\in \ZZ$, then the 
\emph{shift} and \emph{height} of $(i,j+\ell n)$ are defined to be
$$\mathrm{sh}(i, j+\ell n) = \ell
\qquad\hbox{and}\qquad
\mathrm{ht}(i, j+\ell n) = j-i.
$$

Let $\mu = (\mu_1, \ldots, \mu_n)\in \ZZ^n_{\ge 0}$
and let $u_\mu$ be the $n$-periodic permutation defined
by
\begin{equation}
u_\mu(i) = v^{-1}_\mu(i) + n \mu_i , 
\qquad\hbox{for $i\in \{1, \ldots, n\}$.}
\end{equation}
Let $u_\mu(r,c)$ be as defined in \cref{eq:u(rc)}.
Following \cite[Prop. 2.2(a)]{GR21},
the \emph{box-greedy reduced word for $u_\mu$} (an abstract word in symbols $s_1,\ldots, s_{n-1}$ and $\pi)$ is
\begin{equation}
u^\square_\mu = \prod_{\mathrm{boxes}\ (r,c)\ \mathrm{in}\  \mu} (s_{u_\mu(r,c)}\cdots s_2s_1\pi),
\label{bgredwddefn}
\end{equation}
where the product is taken in increasing order of the values $r+nc$.  As in \cite[(2.12)]{GR21},
this reduced word induces a partition of the 
elements of $\mathrm{Inv}(u_\mu)$ according to the boxes $(r,c)$ of $\mu$, and an ordering of the
inversions in each box.
For $(r,c)\in \mu$ and $i\in \{1, \ldots, u_\mu(r,c)\}$,
$$\hbox{the $i$th inversion in the box $(r,c)$ is}\qquad
\beta(i,r,c) = (v_\mu(r), i+n\cdot(\mu_r-c+1)).$$
The inversion $\beta(i,r,c)$ has shift and height
\begin{equation}
\mathrm{sh}(r,c) = \mathrm{sh}(\beta(i,r,c)) = \mu_r-c+1
\quad\hbox{and}\quad
\mathrm{ht}(i,r,c) 
= \mathrm{ht}(\beta(i,r,c)) = v_\mu(r) - i.
\label{rttconversion}
\end{equation}



\begin{example} \label{boxgreedyexample}
If $\mu = (0,4,5,1,4)$ then $n=5$, 
the box-greedy reduced word for $u_\mu$ is
\def\YA{.5}\def\YB{1.5}\def\YC{3}\def\YD{4.5}\def\YE{6}\def\YF{7.9}
\begin{equation}
u^\square_\mu = (s_1\pi)^6(s_2s_1\pi)^7(s_3s_2s_1\pi) = 
\TikZ{[xscale=.8,yscale=-.65]	
\foreach \y in {2,3,4,5}{\node at (1,\y) { $s_1 \pi$};}
\foreach \y in {2, 3}{\node at (2.25,\y) { $s_1 \pi$};}
\foreach \y in {5}{\node at (2.25,\y) { $s_2 s_1 \pi$};}
\foreach \y in {2,3,5}{\node at (3.75,\y) { $s_2 s_1 \pi$};}
\foreach \y in {2,3,5}{\node at (5.25,\y) { $s_2 s_1 \pi$};}
\foreach \y in {3}{\node at (6.95,\y) { $s_3 s_2 s_1 \pi$};}
\draw (\YA,.5) to (\YA,5.5);
\draw (\YB,1.5) to (\YB,5.5);
\draw (\YC,1.5) to (\YC, 3.5) (\YC, 4.5) to (\YC, 5.5);
\draw (\YD,1.5) to (\YD, 3.5) (\YD, 4.5) to (\YD, 5.5);
\draw (\YE,1.5) to (\YE, 3.5) (\YE, 4.5) to (\YE, 5.5);
\draw (\YF, 2.5) to (\YF, 3.5);
\foreach \x [count = \c from 1]  in {\YE, \YF, \YF, \YE, \YE}{
\draw (\YA, \c + .5) to (\x, \c + .5);}
}
\label{bgredwdexample}
\end{equation}
(again multiplying boxes in order by cylindrical numbers) and the inversion set of $u_\mu$ is 
$$
\def\arraystretch{1.1}
\mathrm{Inv}(u_\mu) = 
\text{\setlength{\arraycolsep}{2pt}
\small $\begin{array}{|T|T|T|T|T|}
\multicolumn{1}{|T}{\TikZ{\draw[white] (0,0) to (0,.5);}}
\\\cline{1-4}
\begin{matrix} (3,1+4\cdot 5) \end{matrix}
&\begin{matrix} (3,1+3\cdot 5)  \end{matrix}
& \begin{matrix} (3,1+2\cdot 5) \\ (3,2+2\cdot 5) \end{matrix}
&\begin{matrix} (3, 1+1\cdot 5) \\ (3, 2+1\cdot 5)\end{matrix} 
\\\hline
\begin{matrix} (5,1+5\cdot 5) \end{matrix}
&\begin{matrix} (5,1+4\cdot 5) \end{matrix}\!
&\begin{matrix} (5,1+3\cdot 5) \\ (5, 2 + 3\cdot 5) \end{matrix}
&\begin{matrix} (5, 1+2\cdot 5) \\ (5, 2+2\cdot 5)  \end{matrix}
&\begin{matrix} (5,1+1\cdot 5) \\ (5, 2+1\cdot 5) \\ (5, 3+1\cdot 5) \end{matrix}
\\\hline
\begin{matrix} (2, 1+1\cdot 5)\end{matrix} &\multicolumn{1}{T}{\TikZ{\draw[white] (0,0) to (0,1);}}
\\\cline{1-4}
\begin{matrix} (4, 1+4\cdot 5)\end{matrix}
&\begin{matrix} (4,1+3\cdot 5) \\ (4, 2+3\cdot 5) \end{matrix} 
&\begin{matrix} (4, 1+2\cdot 5) \\ (4, 2 + 2\cdot 5) \end{matrix} 
&\begin{matrix} (4, 1+1\cdot 5) \\ (4, 2+ 1\cdot 5) \end{matrix} \\
\cline{1-4}
\end{array}$}
$$
\end{example}

\subsection{Alcove walks and permutation sequences}

Let $z\in S_n$ and 
choose a subset $F$ of the $s_i$ factors in $u^\square_\mu = w_1w_2\cdots w_r$ to cross out.  
The corresponding \emph{alcove walk of type $(z, u_\mu^\square)$}
is the sequence $p(F)=(p_0, p_1, \ldots, p_r)$ of elements of $W$ 
given by
$$\hbox{$p_0 = z$}\quad\hbox{and}\quad
p_k = \begin{cases}
p_{k-1}\pi, &\hbox{if $w_k = \pi$}, \\
p_{k-1}, &\hbox{if $w_k \in F$ (so that the factor $w_k$ is crossed out)}, \\
p_{k-1}w_k, &\hbox{if $w_k \not\in F$ (so that the factor $w_k$ is not crossed out)}.
\end{cases}
$$
Following \cite[(1.14)]{GR21},
the \emph{permutation sequence of $p(F)$} is the sequence of permutations in $S_n$ given by
$$\overline{p}(F) = (\overline{p_0}, \overline{p_1}, \ldots, \overline{p_r}),
\qquad\hbox{where}\quad
\overline{w}(i) = w(i) \bmod n.
$$
Note that $\overline{w_1w_2} = \overline{w_1}\,\overline{w_2}$ and 
$\overline{\pi} = c_n = s_1\cdots s_{n-1}$ and
$\overline{s_j} = s_j$ for $j\in \{1, \ldots, n-1\}$.

\subsection{Converting to set-valued tableaux} \label{sec:conversion}

The bijection between alcove walks and set-valued tableaux is
$$
\begin{matrix}
\{\hbox{alcove walks of type $u^\square_\mu$}\} &\longleftrightarrow &\{\hbox{set-valued tableaux of shape $\mu$}\} \\
p(F) &\longmapsto &T
\end{matrix}
$$
where, in the set-valued tableau $T$ corresponding to the alcove walk $p(F)$,
the set $T(r,c)$ in box $(r,c)$ specifies which factors are crossed out in that box: if a box in $T$ contains $i$, then 
delete $s_i$ in the corresponding word. 


\begin{example} \label{pfbyexample}
In the case $\mu = (0,4,5,1,4)$, where the box-greedy reduced word for $u_\mu$ is as given in 
\eqref{bgredwdexample},
there are $6+2\cdot 7+3=23$ factors of the form $s_i$ in $u^\square_\mu$ and 
so there are a total of $2^{23}$ alcove walks of type $(z,u^\square_\mu)$ 
(for any fixed permutation $z\in S_n$).
Each alcove walk corresponds to a choice of the $s_i$ factors in $u^\square_\mu$ to cross out.  

The tableau $T$ in \cref{runningexample} corresponds to the subset 
\def\YA{.5}\def\YB{1.5}\def\YC{3}\def\YD{4.5}\def\YE{6}\def\YF{8}
\begin{equation}
F = 
\TikZ{[xscale=.8,yscale=-.7]	
\foreach \y in {2,3}{\node at (1,\y) {\XWord{1}{1}};}
\foreach \y in {4,5}{\node at (1,\y) { $s_1\pi$};}
\foreach \y in {2,3}{\node at (2.25,\y) { $s_1 \pi$};}
\foreach \y in {5}{\node at (2.25,\y) { $s_2 s_1 \pi$};}
\foreach \y in {2}{\node at (3.75,\y) {\XWord{2}{1}};}
\foreach \y in {3,5}{\node at (3.75,\y) {\XWord{2}{1,2}};}
\foreach \y in {2}{\node at (5.25,\y) {\XWord{2}{2}};}
\foreach \y in {3,5}{\node at (5.25,\y) { $s_2 s_1 \pi$};}
\foreach \y in {3}{\node at (7,\y) {\XWord{3}{1,3}};}
\draw (\YA,.5) to (\YA,5.5);
\draw (\YB,1.5) to (\YB,5.5);
\draw (\YC,1.5) to (\YC, 3.5) (\YC, 4.5) to (\YC, 5.5);
\draw (\YD,1.5) to (\YD, 3.5) (\YD, 4.5) to (\YD, 5.5);
\draw (\YE,1.5) to (\YE, 3.5) (\YE, 4.5) to (\YE, 5.5);
\draw (\YF, 2.5) to (\YF, 3.5);
\foreach \x [count = \c from 1]  in {\YE, \YF, \YF, \YE, \YE}{
\draw (\YA, \c + .5) to (\x, \c + .5);}
}\ ,
\label{alcwalkexample}
\end{equation}
which has alcove walk
$$p(F)=(p_0, p_1, \ldots, p_{37}) = (z, z, z\pi, z\pi, z\pi^2, z\pi^2 s_1, z\pi^2 s_1 \pi, z\pi^2 s_1 \pi s_1, z\pi^2 s_1 \pi s_1 \pi,   \ldots)$$
(there is a repeat entry in $p(F)$ each time there is an $s_i$ crossed out in $F$).
Using one line notation $w=(w(1)\ w(2)\ \cdots w(n))$ for $n$-periodic permutations (underlining 2-digit terms for emphasis), a box-by-box formulation in the case that $z = \id = (12345)$ is
{\def\arraystretch{1.2}
\setlength{\arraycolsep}{3pt}
\begin{equation*}
p(F) = \quad \text{\small$
\begin{array}{|T|T|T|T|T|}
\multicolumn{1}{|T}{\TikZ{\draw[white] (0,0) to (0,.75);}}
\\\cline{1-4}
\begin{matrix} (1\ 2\ 3\ 4\ 5) \\ (2\ 3\ 4\ 5\ 6)  \end{matrix} 
&\begin{matrix}  (6\ 3\ 7\ 9\ \underline{10}) \\ (3\ 7\ 9\ \underline{10}\ \uoverline{11}) \end{matrix}
&\begin{matrix} 
(3\ \underline{11}\ 9\ \uoverline{12}\ \underline{15}) \\
(3\ \underline{11}\ 9\ \uoverline{12}\ \underline{15}) \\
(\underline{11}\ 9\ \uoverline{12}\ \underline{15}\ 8) 
\end{matrix}
&\begin{matrix}
(\uoverline{12}\ \underline{15}\ 8\ \uoverline{16}\ \underline{14}) \\
(\uoverline{15}\ \underline{12}\ 8\ \uoverline{16}\ \underline{14}) \\
(\underline{12}\ 8\ \uoverline{16}\ \underline{14}\ \uoverline{20})
\end{matrix}
\\\hline
\begin{matrix} (2\ 3\ 4\ 5\ 6) \\ (3\ 4\ 5\ 6\ 7) \end{matrix} 
&\begin{matrix}   (7\ 3\ 9\ \underline{10}\ \uoverline{11}) \\ 
(3\ 9\ \underline{10}\ \uoverline{11}\ \underline{12})  \end{matrix}
&\begin{matrix} 
(\underline{11}\ 9\ \uoverline{12}\ \underline{15}\ 8) \\
(\underline{11}\ 9\ \uoverline{12}\ \underline{15}\ 8) \\
(9\ \uoverline{12}\ \underline{15}\ 8\ \underline{16}) 
\end{matrix}
&\begin{matrix} 
(\underline{12}\ \uoverline{16}\ 8\ \underline{14}\ \uoverline{20}) \\
(\underline{16}\ \uoverline{12}\ 8\ \underline{14}\ \uoverline{20}) \\
(\uoverline{12}8\underline{14}\ \uoverline{20}\ \underline{21})
\end{matrix}
&\begin{matrix} 
(\underline{12}\ 8\ \uoverline{20}\ \underline{21}\ \uoverline{19}) \\
(\underline{12}\ \uoverline{20}\ 8\ \underline{21}\ \uoverline{19}) \\
(\underline{12}\ \uoverline{20}\ 8\ \underline{21}\ \uoverline{19}) \\
(\uoverline{20}\ 8\ \underline{21}\ \uoverline{19}\ \underline{17})
\end{matrix} 
\\\hline
\begin{matrix} (4\ 3\ 5\ 6\ 7) \\ (3\ 5\ 6\ 7\ 9)\end{matrix} 
\\\cline{1-4}
\begin{matrix} (5\ 3\ 6\ 7\ 9) \\ (3\ 6\ 7\ 9\ \underline{10}) \end{matrix}
&\begin{matrix} (3\ \underline{10}\ 9\ \uoverline{11}\ \underline{12}) \\
(\underline{10}\ 3\ 9\ \uoverline{11}\ \underline{12}) \\ 
(3\ 9\ \underline{11}\ \uoverline{12}\ \underline{15})
\end{matrix}
&\begin{matrix} 
(9\ \uoverline{12}\ \underline{15}\ 8\ \uoverline{16}) \\
(9\ \uoverline{12}\ \underline{15}\ 8\ \uoverline{16}) \\
(\uoverline{12}\ \underline{15}\ 8\ \uoverline{16}\ \underline{14}) 
\end{matrix}
&\begin{matrix} 
(\uoverline{12}\ \underline{14}\ 8\ \uoverline{20}\ \underline{21}) \\
(\uoverline{14}\ \underline{12}\ 8\ \uoverline{20}\ \underline{21}) \\
(\underline{12}\ 8\ \uoverline{20}\ \underline{21}\ \uoverline{19})
\end{matrix}\\\cline{1-4}
\end{array}
$}
\end{equation*}}
The permutation sequence is
$$\overline{p}(F)=(\overline{p_0}, \overline{p_1}, \ldots, \overline{p_{37}}) 
	= (z, z, zc_n, zc_n, zc_n^2, zc_n^2 s_1, zc_n^2 s_1 c_n, zc_n^2 s_1 c_n s_1, zc_n^2 s_1 c_n s_1 c_n,   \ldots)$$
(obtained by replacing each $\pi$ by the $n$-cycle $c_n = s_1\cdots s_{n-1} = \gamma_n^{-1}$).
Namely, $\overline{p}(F)$ is obtained by reducing all the values in $p(F)$ mod $n$ (and $n=5$ for $\mu = (0,4,1,5,4)$).
\begin{equation}
\overline{p}(F) = \quad 
\text{\small
\setlength{\arraycolsep}{3pt}
$
\begin{array}{|T|T|T|T|T|}
\multicolumn{1}{|T}{\TikZ{\draw[white] (0,0) to (0,.75);}}
\\\cline{1-4}
\begin{matrix} (\blue{\bf 12}345) \\ (23451)  \end{matrix} 
&\begin{matrix}  (13245) \\ (32451) \end{matrix}
&\begin{matrix} (31425) \\
(\blue{\bf 31}425) \\
(14253) 
\end{matrix}
&\begin{matrix}
(2\blue{\bf 53}14)  \\
(52314)  \\
(23145)  
\end{matrix}
\\\hline
\begin{matrix} (\blue{\bf 23}451) \\ (34512)  \end{matrix} 
&\begin{matrix}   (23451) \\ 
(34512)  \end{matrix}
&\begin{matrix} 
(1\blue{\bf 42}53)  \\
(\blue{\bf 14}253)  \\
(42531)  
\end{matrix}
&\begin{matrix} 
(21345)  \\
(12345)  \\
(24351) 
\end{matrix}
&\begin{matrix} 
(23\blue{\bf 51}4)  \\
(25314)  \\
(\blue{\bf 25}314)  \\
(53142)  \\
\end{matrix} 
\\\hline
\begin{matrix} (43512) \\ (35124) \end{matrix} 
\\
\cline{1-4}
\begin{matrix} (53124) \\ (31245) \end{matrix}
&\begin{matrix} (35412) \\
(53412) \\ 
(34125)
\end{matrix}
&\begin{matrix} 
(4\blue{\bf 25}31)  \\
(\blue{\bf 42}531)  \\
(25314)  
\end{matrix}
&\begin{matrix} 
(24351)  \\
(42351)  \\
(23514)  
\end{matrix}\\\cline{1-4}
\end{array}
$}
\label{alcwalkexample}
\end{equation}
The \emph{box-by-box permutation sequence} 
of $F$ is obtained by recording the last permutation in each box, which is $z^T$ in \eqref{eq:zFavT}. 
The elements $z^T_{(r,c)}$ in each box of $z^T$ are also obtained from $z=z^T_{\mathrm{init}}$ and $T$ by
the formula in \cref{zTrcdefn}. 

We have used the blue highlighted numbers in \eqref{alcwalkexample}
to record the positions $i$, $i+1$ when there is a repeat entry
in the sequence $\overline{p}(F)$ coming from a crossed out $s_i$.  
These record the information of $F$.  This same information is translated into the underlines in $z^T$ 
in \eqref{eq:zFavT}, and the underlines exactly specify the entries of $T$.
The set-valued tableau indicating the positions of the underlined numbers in each box is the tableau in \eqref{eq:favoriteT}.
\end{example}


\subsection{Conversion of the statistics}

The proof of \cref{pospowers} is completed by matching up the statistics in \cref{pospowers}
with the statistics that appear in \cite[Theorem 3.1]{RY08}.  In \cref{pfbyexample}, we have used the 
blue highlighted numbers in \eqref{alcwalkexample} and the underlines in \eqref{eq:zFavT} 
to illustrate how the information of 
the `folds' in \cite[Theorem 3.1]{RY08} is equivalent to the information of the set-valued tableau $T$.
In the context of \cite[(2.36)]{RY08}, the set $T(r,c)$ exactly records the folds coming from the box $(r,c)$.
By \eqref{rttconversion},
the factor
$\prod_{(r,c)\in \mu} \prod_{m\in T(r,c)} \frac{(1-t)}{1-q^{\mathrm{sh}(r,c)}t^{\mathrm{ht}(i,r,c)}}$
which appears in the product in \cref{pospowers} corresponds to
$$
\hbox{the factors}\quad
\left(\prod_{k\in f^+(p)} \frac{(1-t_{\beta^\vee_k})}{1-q^{\langle -\beta^\vee_k, \rho_c\rangle}} \right)
\left(\prod_{k\in f^-(p)} \frac{(1-t_{\beta^\vee_k})}{1-q^{\langle -\beta^\vee_k, \rho_c\rangle}} \right)
\quad
\hbox{in \ \cite[Theorem 3.1]{RY08}.}
$$
The sets $T^z_{<}$ and $T^z_{>}$ correspond to the sets $f^+(p)$ and $f^-(p)$ of positive and negative folds in \cite[Theorem 3.1]{RY08}.
The permutation $z^T_{\mathrm{fin}}$ is the permutation denoted $\varphi(p)$ in \cite[Theorem 3.1]{RY08}
(denoted $z_r$ in \cite[(1.14)]{GR21}),
and the permutation $z^T_{\mathrm{init}}v_\mu^{-1}$ is the permutation $m$ of \cite[Theorem 3.1]{RY08}.
With these conversions, the last term 
$\hbox{$\frac12$} \big(\ \ell(z^T_{\mathrm{fin}}) - \ell(z^T_{\mathrm{init}}v_\mu^{-1}) - \vert T\vert \ \big)$
in the definition of $\mathrm{cov}_>^z(T)$ in \cref{pnfolds}
corresponds to the combination of the 
factors $t^{\frac12}_{\varphi(p)}$ and $t^{-\frac12}_{\beta^\vee_k}$  in \cite[Theorem 3.1]{RY08} 
and the factor $t^{\frac12}_m$ in \cite[Remark 3.2]{RY08}.



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