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\title{A combinatorial formula for Macdonald cumulants}

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%% use \addressmark{1}, \addressmark{2} etc for the institutions, and
%% use \thanks{} for contact details
\author{Maciej
Do{\l}\k{e}ga\addressmark{1,2}\thanks{\href{mailto:maciej.dolega@amu.edu.pl}{maciej.dolega@amu.edu.pl}.
Acknowledges support from {\it Narodowe Centrum Nauki}, grant
UMO-2015/16/S/ST1/00420.}}
 \address{\addressmark{1}Wydzia{\l} Matematyki i Informatyki, 
Uniwersytet im.~Adama Mickiewicza, 
Umultowska 87, 
61-614 Pozna{\'n}, 
Poland, \\ 
\addressmark{2}Instytut Matematyczny,
Uniwersytet Wroc{\l}awski,  \mbox{pl.\ Grunwaldzki~2/4,} 50-384
Wroc{\l}aw, Poland}

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\received{\today}

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%% put your English abstract here, or comment this out if you don't have one yet
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\abstract{Macdonald cumulants are symmetric functions
  that generalize Macdonald polynomials. We prove a combinatorial formula for them which
extends the celebrated formula of Haglund for Macdonald
polynomials. We also provide several applications of our formula -- it gives a new, constructive
proof of a strong factorization property
of Macdonald polynomials and it proves that Macdonald cumulants
are $q,t$--positive in the monomial and in the fundamental
quasisymmetric bases. Furthermore, we use our formula to prove the recent higher-order
Macdonald positivity conjecture for the coefficients
of the Schur polynomials indexed by hooks. Our combinatorial formula
links Macdonald cumulants to $G$--parking
functions of Postnikov and Shapiro.}

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% \resume{}

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\keywords{Macdonald polynomials; Schur polynomials; Cumulants; Tutte polynomials; $G$--Parking
  functions; $q,t$-Kostka numbers}

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\begin{document}

\maketitle
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\section{Macdonald cumulants}
\label{sec:introduction}

The main character in this paper is a certain symmetric function
$\ka(\la^1,\dots,\la^r)(\xx;q,t)$ indexed by a finite set of
partitions $\{\la^1,\dots,\la^r\}$ and two additional parameters
$q,t$. This function is called \emph{Macdonald cumulant} and this
extended abstract describes the main ideas presented in the sequence of papers
\cite{DolegaFeray2017,Dolega2017,Dolega2017b} which initiated study of
Macdonald cumulants.

\subsection{Motivation}

The initial motivation for studying cumulants $\ka(\la_1,\dots,
\la_r)$ comes from our attempts \cite{DolegaFeray2017} on proving the $b$-conjecture
-- one of the major open problems in the theory of Jack symmetric
functions posed by Goulden and Jackson \cite{GouldenJackson1996}.
The $b$-conjecture states that the coefficients of a certain multivariate
generating function $\psi(\xx,\yy,\zz;\beta)$ involving Jack symmetric functions can be
interpreted as weighted generating functions of graphs embedded into
surfaces. Except some special cases
\cite{LaCroix2009,
  Dolega2017a} not much is known and the $b$-conjecture is still wide
open. However, in our recent paper \cite{DolegaFeray2017} the author and F\'eray were able to
rewrite the function $\psi(\xx,\yy,\zz;\beta)$ as a linear combination
of cumulants of Jack symmetric functions, which are specializations of
$\ka(\la_1,\dots,
\la_r)$. In view of this result, understanding of the structure of
Macdonald cumulants is of great interest as a potential tool for solving
the $b$-conjecture.

Let $\A$ be a commutative ring with two different multiplicative
structures $\cdot$
and $\oplus$ which define two (different) algebra structures
on $\A$. Then, for any $X_1,\dots,X_r \in \A$ one can define a \emph{conditional
cumulant} $\ka(X_1,\dots,X_r) \in \A$ by 
the following generating series
\begin{equation} 
\label{eq:cumu1}
\ka(X_1,\dots,X_r) := [t_1\cdots t_r] \log\E
  e^{t_1X_1+\cdots t_rX_r},
\end{equation}
where $\E: (\A,\oplus) \to (\A,\cdot)$ is the identity map. This cumulant
has the following interpretation: it measures the discrepancy between those two algebraic structures.
Definition \eqref{eq:cumu1} can be transformed into an equivalent
but more
combinatorial definition:
\begin{equation}
\label{eq:DefCumulants} 
\ka(X_1,\dots,X_r) = \sum_{\substack{\pi \in \PPP([r]) } } 
            (-1)^{\#\pi-1}(\#\pi-1)! \prod_{B \in \pi} u_B,
\end{equation}
where $u_B := \E\left(\bigoplus_{b \in B}X_b\right) = \bigoplus_{b \in B}X_b$
and we sum over \emph{set-partitions} of $[r] := \{1,2,\dots,r\}$, that is all possible
sets $\pi$ of
nonempty subsets of $[r]$ such that every element $i \in [r]$ belongs
to precisely one element of $\pi$; here $\#\pi$ denotes the number of elements
of $\pi$.

We go back to the symmetric functions now. A classical problem in the
symmetric functions theory is to understand the \emph{structure
  constants} $a_{\mu,\nu}^\la $ of a given linear basis $\{s_\mu\}_\mu$:
\[ s_\mu\cdot s_\nu = \sum_{\la} a_{\mu,\nu}^\la\ s_\la.\]
The celebrated basis of Macdonald polynomials $\left\{\tilde{H}_\mu(\xx;q,t)\right\}_\mu$ (here, we use ``transformed form'' of Macdonald
polynomials sometimes called ``modified form''; see
\cite{Haiman2001} for more details) is an interesting example. It was shown by Macdonald \cite{Macdonald1995} (and it is
straightforward from Haglund's formula \eqref{eq:Haglund}) that one has the following structure
constants in the specialization $q=1$:
\[ \tilde{H}_\mu(\xx;1,t)\cdot \tilde{H}_\nu(\xx;1,t) =
  \tilde{H}_{\mu \oplus \nu}(\xx;1,t),\]
where for partitions $\lambda = (\lambda_1,\lambda_2,\dots)$ and $\mu =
(\mu_1,\mu_2,\dots)$ we define a new partition $\la\oplus\mu :=
(\la_1+\mu_1,\la_2+\mu_2,\dots)$ by adding
coordinates of partitions $\la$ and $\mu$.
Since Macdonald polynomials $\{\tilde{H}_\mu\}_\mu$ form a linear
basis of the algebra $\Lambda$ of symmetric functions over $\QQ(q,t)$,
we can define a new multiplicative structure $(\Lambda,\oplus)$ by setting $\tilde{H}_\mu
\oplus \tilde{H}_\nu := \tilde{H}_{\mu\oplus\nu}$ and extending it by
linearity.
Therefore $(\Lambda,\oplus)$ can be
interpreted as an approximation of the algebra $(\Lambda,\cdot)$ of
interest, as $q \to 1$ and it is natural to study cumulants of the form
$\ka(f_1,\dots,f_r)$, where $f_1,\dots,f_r \in \Lambda$, which describe this
approximation on the ``higher--order'' level and may bring a better
understanding of the structure constants for Macdonald polynomials. By multilinearity of
cumulants it is enough to study $\ka(\tilde{H}_{\la_1},\dots,
\tilde{H}_{\la_r})$ and it was first conjectured in
\cite{DolegaFeray2017}, and then proved in \cite{Dolega2017}, that this
``higher--order'' level approximation, is indeed of an expected order:
\begin{theorem}[\cite{Dolega2017}]
\label{theo:podzielnosc}
For any partitions $\lambda^1,\dots,\lambda^r$ one has
\[ \ka(\tilde{H}_{\la^1},\dots,
\tilde{H}_{\la^r}) \in \Z[q,t]\{(q-1)^{r-1}m_\mu\}_\mu.\]
\end{theorem}
It is therefore natural to introduce the \emph{Macdonald cumulant}
$\ka(\la^1,\dots,\la^r)(\xx;q,t)$ as
\begin{equation}
\label{eq:MacdoCumu}
\ka(\la^1,\dots,\la^r)(\xx;q,t) := \frac{\ka(\tilde{H}_{\la_1}(\xx;q,t),\dots,
\tilde{H}_{\la_r}(\xx;q,t))}{(q-1)^{r-1}}.
\end{equation}

We finish this section by mentioning one more natural motivation for
studying Macdonald cumulants. One of the most typical
application of cumulants is to show that a certain family of random
variables is asymptotically Gaussian. The main
technique is to show that conditional cumulants of ``observables''
have a certain \emph{small cumulant
  property} exactly of the same form as in \cref{theo:podzielnosc}; see \cite{Sniady2006c, FerayMeliot2012}. It is therefore natural to ask for a
probabilistic interpretation of \cref{theo:podzielnosc}, which leads
to some kind of a central limit theorem. The most natural framework
to investigate this problem seems to be related with Macdonald
processes introduced by Borodin and Corwin \cite{BorodinCorwin2014}
and we leave this problem for the future research.

\subsection{Schur--positivity problem}
\label{subsec:SchurPositivity}

Our personal favorite reason for studying cumulants $\ka(\la^1,\dots,\la^r)$ is their beautiful and mysterious
combinatorial structure. We recall that monomial symmetric functions have integer coefficients in the Schur
basis expansion, thus one can reformulate \cref{theo:podzielnosc} as
follows: for any partitions $\la^1,\dots,\la^r$ one has the following expansion
\[ \ka(\la^1,\dots,\la^r) \in \Z[q,t]\{s_\mu\}_\mu.\]
Remarkably, extensive computer simulations suggest that Macdonald
cumulants are, in fact, Schur--positive, which we conjectured in our
recent paper \cite{Dolega2017}:


\begin{conjecture}[Higher--order Macdonald positivity conjecture \cite{Dolega2017}]
\label{conj:GeneralizedKostka}
Let $\lambda^1,\dots,\lambda^r$ be partitions. Then, for any
partition $\mu$, the \emph{multivariate $q,t$-Kostka number}
$\tilde{K}^{(q,t)}_{\mu; \la^1,\dots,\la^r}$ defined by the following
expansion
\[ \ka(\la^1,\dots,\la^r) :=
  \sum_{\mu}\tilde{K}^{(q,t)}_{\mu; \la^1,\dots,\la^r}\ s_\mu\]  
is a polynomial in $q,t$ with
\textbf{nonnegative integer} coefficients.
\end{conjecture}

Note that the special case $r=1$ of \cref{conj:GeneralizedKostka} corresponds to the Macdonald positivity
ex-conjecture \cite{Macdonald1995} which says that the coefficients
$\tilde{K}_{\lambda,\mu}(q,t)$ in the following expansion:
\[ \tilde{H}_\mu(\bm{x};q,t) =
  \sum_{\lambda}\tilde{K}_{\lambda,\mu}(q,t)\ s_\la(\xx)\]
are polynomials in $q,t$ with
nonnegative integer coefficients. It took more than
ten years to Haiman to prove it \cite{Haiman2001} by connecting
their conjectural representation theoretic interpretation with the problem from algebraic geometry and solving it. This result is considered as a
great breakthrough in the symmetric functions theory and it initiated
very active research in the remarkable algebraic combinatorics of the
Macdonald polynomials. Thus, our conjecture generalizes Macdonald positivity
ex-conjecture from the cumulant of
order $1$ to cumulants of higher order.  

The main result of this paper is an explicit combinatorial formula for Macdonald
cumulants $\ka(\la^1,\dots,\la^r)$ and its applications. Moreover, it will
be shortly clear that our formula connects Macdonald cumulants with some
seemingly unrelated topics from algebraic combinatorics. We believe
that these links enrich the world of
combinatorics of
Macdonald polynomials and give one more motivation for studying the
new interesting family of the symmetric functions $\{\ka(\la^1,\la^2,\dots)\}_{\la^1,\la^2,\dots}$.

\section{$G$--parking functions and fillings of the Young diagrams}

Before we describe our formula, let us briefly recall the
celebrated combinatorial formula for Macdonald polynomials, which
coincides with our formula for a trivial cumulant $\ka(\la) =
\tilde{H}_{\la_r}(\xx;q,t)$ and which was a great source of
inspiration for our work.

\begin{theorem}[\cite{HaglundHaimanLoehr2005}]
\label{theo:Haglund}
Let $\la$ be a partition. Then
\begin{equation}
\label{eq:Haglund}
\tilde{H}_{\la}(\bm{x};q,t) = \sum_{\si:\la\to\N_+}q^{\inv(\si)}\
t^{\maj(\si)}\ \xx^\si,
\end{equation}
where we sum over all possible fillings of $\la$ by nonnegative
integers, and where $\xx^\si:=\prod_{\square \in \la}x_{\si(\square)}$.
\end{theorem}

With each filling $\si: \la \to \N_+$ one can associate certain
statistics $\inv(\si)$ and $\maj(\si)$ which play
crucial role in \eqref{eq:Haglund} and we are
going to describe them in the following. 

\subsection{Fillings of Young diagrams and their statistics}
\label{subsec:Fillings}

For any partition $\la \vdash n$ let $\si:\lambda \to \N_+$ be a filling of the boxes of
the diagram $\la$ by positive integers. A \emph{descent} of $\si$ is a
pair of entries $\si(\square) > \si(\square')$ such that $\square$ lies immediately
above $\square'$. Let $\ell_\la(\square)$ denotes the number of boxes in $\la$ lying
in the same column as $\square$ strictly above it. The \emph{major index} $\maj(\si)$ of a filling
$\si$ is defined as:
\begin{equation}
\label{eq:major}
\maj(\si) := \sum_{\square \in \Des(\si)}(\ell_\la(\square)+1),
\end{equation}
where $\Des(\si) := \{\square \in \la: \si(\square) > \si(\square')
\text{ is a descent} \}$ is a \emph{set of descents}.

\begin{figure}[t]
        \[
          \begin{array}{c}
        \YFrench
    \Yboxdim{1cm}
    \begin{tikzpicture}[scale=0.6]
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        \newcommand\eleven{11}
        \tgyoung(0cm,0cm,12<10>\eleven246<12><14>\eleven
        <13>,!\yg 24!\yw 31!\yg 8<10>7!\yw 899,!\yg 674<10>9!\yw 9!\yg
        <13><13>,!\yw 111!\yg \eleven!\yw 444,!\yg 934!\yw 9)
       \Ylinethick{1.5pt}
       % \Yfillopacity{0}
       % \tgyoung(0cm,1cm,;;::;;;:,;;;;;:;;,:::;,;;;)
        \draw[gray, thick] (0.5,4.5) to (1.5,4.5);
        \draw[gray, thick] (0.5,4.5) to [bend left] (2.5,4.5);
        \draw[gray, thick] (0.5,3.5) to [bend left] (3.5,3.5);
        \draw[gray, thick] (1.5,3.5) to [bend left] (3.5,3.5);
        \draw[gray, thick] (2.5,3.5) to (3.5,3.5);
        \draw[gray, thick] (0.5,3.5) to [bend left] (3.5,3.5);
        \draw[gray, thick] (0.5,2.5) to [bend left] (2.5,2.5);
        \draw[gray, thick] (3.5,2.5) to (4.5,2.5);
        \draw[gray, thick] (3.5,2.5) to [bend left] (5.5,2.5);
        \draw[gray, thick] (1.5,1.5) to (2.5,1.5);
        \draw[gray, thick] (2.5,1.5) to (3.5,1.5);
        \draw[gray, thick] (4.5,1.5) to [bend left]  (6.5,1.5);
        \draw[gray, thick] (5.5,1.5) to (6.5,1.5);
        \draw[gray, thick] (5.5,1.5) to [bend left] (7.5,1.5);
        \draw[gray, thick] (5.5,1.5) to [bend left] (8.5,1.5);
        \draw[gray, thick] (5.5,1.5) to [bend left] (9.5,1.5);
        \draw[gray, thick] (2.5,0.5) to [bend left] (4.5,0.5);
        \draw[gray, thick] (2.5,0.5) to [bend left] (5.5,0.5);
        \draw[gray, thick] (2.5,0.5) to [bend left] (6.5,0.5);
        \draw[gray, thick] (3.5,0.5) to (4.5,0.5);
        \draw[gray, thick] (3.5,0.5) to [bend left] (5.5,0.5);
        \draw[gray, thick] (3.5,0.5) to [bend left] (6.5,0.5);
        \draw[gray, thick] (7.5,0.5) to [bend left] (9.5,0.5);
        \draw[gray, thick] (8.5,0.5) to (9.5,0.5);
        \draw[gray, thick] (8.5,0.5) to [bend left] (10.5,0.5);




       \end{tikzpicture}
\end{array}
        \]
        \caption{Inversion pairs in the above filling $\si$ are indicated by
          grey lines, while the set of descents is highlighted in light
          gray.}
        \label{fig:LegArmSum}
    \end{figure}

% The second statistic that is of great importance in this paper is
% a certain generalization of inversions in a permutation. First, we say
% that two boxes $\square,\square' \in \la$ \emph{attack each other} if
% either
% \begin{itemize}
% \item
% they are in the same row: $\square = (i,j), \square' = (k,j)$, or;
% \item
% they are in consecutive rows, with the box in the upper row strictly to the right of
% the one in the lower row: $\square = (i, j+1), \square' = (k, j)$, where $i > k$.
% \end{itemize} 

We also define the set of \emph{inversion pairs}
$(\square_1,\square_2)$, where $\square_1$ is a box lying in the same
row as $\square_2$ to its left, $\square_3$ is a box lying directly below $\square_1$ and such that
$\square_1,\square_2,\square_3$ are \emph{counterclockwise
  increasing}. That is, one of the following conditions holds true:
$\si(\square_1) \leq \si(\square_3) < \si(\square_2)$, or
$\si(\square_3) < \si(\square_2) < \si(\square_1)$, or $\si(\square_2)
< \si(\square_1) \leq \si(\square_3)$. Here, the convention is that for $\square_1,\square_2$ lying in the
first row $\si(\square_3) < \min_{\square \in \la}{\si(\square)}$.
The set of inversion pairs of $\si$ is denoted by
$\InvP(\si)$. \cref{fig:LegArmSum} presents an example of above
defined objects. We set
\begin{equation}
\label{eq:inv}
\inv(\si) := \#\InvP(\si).
\end{equation}

Finally, the \emph{reading order} is the linear ordering of the entries of $\la$ given by reading them row by row,
top to bottom, and left to right within each row. With any filling
$\si$ we associate its \emph{reading word} $w_\si$ by reading
its entries in the reading order. 

\subsection{$G$-parking functions and the abelian sandpile model}
\label{subsec:GParking}

Let $G = (V,E)$ be a rooted, connected multigraph (multiple edges and
loops are allowed) with the set of vertices $V = [r]$, where $r\geq 1$ is a positive
integer and we denote the root of $G$ by $v \in [r]$. For
any $i \in U \subset [r]\setminus\{v\}$ we define the \emph{outdegree}
$\outdeg_U(i)$ of a vertex $i$ as the number of edges in $G$ linking $i$
with some vertex $j \notin U$. We call a function $f: [r]\setminus\{v\} \to \N$ a \emph{$G$-parking function} if for any nonempty subset $U \subset [r]\setminus\{v\}$ there exists $i \in U$
such that $f(i) < \outdeg_U(i)$.

When $G = K_{r}$ is the complete graph on $[r]$, then the set of $G$-parking functions is
precisely the set of parking functions. 

Postnikov and Shapiro noticed that $G$-parking functions are directly
related to \emph{recurrent configurations}
in the \emph{abelian sandpile model for $G$}, which is a model where we are
trying to distribute chips among vertices of our graph. A function $u : [r]\setminus\{v\} \to
\N$ giving the number of chips placed in vertices of $G$ different
from the root is called a \emph{configuration}. We say that a vertex $i \in [r]\setminus\{v\}$ is \emph{unstable} if $u(i)
\geq \deg(i)$ -- if this is a case, this vertex can \emph{topple} by
sending chips to adjacent vertices one along each incident
edge. We say that a configuration is \emph{stable} if all the
vertices $i \in [r]\setminus\{v\}$ except the root are stable. For the root we set
$u(v) = -\sum_{i \in [r]\setminus\{v\}}u(i)$, and the root can always topple. Finally, we say that a
configuration $u$ is \emph{recurrent} if there exists a nontrivial
configuration $u' \neq 0$ such that $u$ can be obtained from $u+u'$ by a sequence
of topplings. Postnikov and Shapiro noticed that a configuration $u$ is
recurrent if and only if $f:[r]\setminus\{v\}\to\N$ defined by $f(i):=
\deg(i)-u(i)-1$ is a $G$-parking function. 
We define a \emph{weight} of a $G$-parking function $f$:
\[ \wt(f) := \#E-r-\sum_{i \in [r]\setminus\{v\}}f(i),\]
and we associate a $q$-generating function of $G$-parking functions $\I_G(q) := \sum_{f}q^{\wt(f)}$.

Merino L\'opez proved \cite{MerinoLopez1997} that $\I_G(q) = \Tu_G(1,q)$,
where $\Tu_G(t,q)$ is a Tutte polynomial of $G = (V,E)$, that is
\begin{equation}
\label{eq:DefTutte}
T_G(x,y) = \sum_{H \subset G}(x-1)^{c(H)-1}\ (y-1)^{\#E(H)-\#V+c(H)},
\end{equation}
where we sum over all (possibly disconnected) sub-multigraphs of $G$, $c(H)$ denotes the
number of connected components of $H$, and $E(H)$ is the set of edges
of $H$. Therefore in the special case
of the complete graph $\I_G(q)$ is the inversion polynomial (the
generating function of rooted trees with respect to
the number of their inversions) and in general we call it \emph{$G$--inversion polynomial}.

For technical reasons, we extend the definition of $\I_G(q)$ to the set of
all finite multigraphs by setting $\I_G(q)=0$ when $G$ is
not connected.

\subsection{Combinatorial formula for Macdonald cumulants}

We are ready to formulate our main result. 
% We begin with the
% combinatorial formula for Macdonald cumulants, which generalizes the
% celebrated Haglund's formula \cite{HaglundHaimanLoehr2005}:
\begin{theorem}
\label{theo:MacdonaldCumuFormula}
Let $\la^1,\dots,\la^r$ be partitions. Then, the following formula
holds true:
\begin{equation}
\label{eq:MacdonaldCumuFormula}
\ka(\la^1,\dots,\la^r) = \sum_{\si: \la^{[r]} \to
  \N_+} \I_{G^\si_{\la^1,\dots,\la^r}}(q)\ t^{\maj(\sigma)}\ \xx^\sigma.
\end{equation}
\end{theorem}

The summation index in \eqref{eq:MacdonaldCumuFormula} runs over all
the fillings $\sigma$ of a Young diagram $\la^{[r]}$ by positive
integers, where $\la^B := \bigoplus_{b \in B}\la^b$ and $G^{\sigma}_{\la^1,\dots,\la^r}$ is a certain multigraph
and its construction is presented in the following section.

\subsection{Consequences of the formula}

Since our combinatorial formula for Macdonald cumulants is a
generalization of \eqref{eq:Haglund}, it has similar implications. Let us briefly
summarize them, before we go into details of a
description of $G^{\sigma}_{\la^1,\dots,\la^r}$.

Our main result extends \cref{theo:podzielnosc} twofold. Firstly, our
formula shows that the coefficients of Macdonald cumulants in the monomial expansion are
positive integers which is a stronger statement. Secondly, the original proof of
\cref{theo:podzielnosc} relies on some abstract arguments and
complicated induction -- in
particular it does not tell anything about the combinatorial
structure of Macdonald cumulants, contrary to our formula which
provides an explicit combinatorial interpretation in terms of $G$--parking functions. 

Moreover, we deduce from our formula an explicit expansion of
Macdonald cumulants in the fundamental quasisymmetric
functions, which turns out to be $q,t$--positive.

Finally, our main result implies that \cref{conj:GeneralizedKostka} holds true in the
special case of hooks. 

We describe these applications in \cref{sec:Fundamental}.

\section{Coloring of the Young diagram $\la^{[r]}$ and multigraphs}
\label{sec:Coloring}

\subsection{Coloring of the Young diagram $\la^{[r]}$}
\label{subsect:Identification}

    % \begin{figure}[t]
    %     \[
    %       \begin{array}{c}
    %     \newcommand\yb{\Yfillcolour{cyan!70}}
    %     \newcommand\yr{\Yfillcolour{BrickRed}}
    %     \newcommand\yw{\Yfillcolour{white}}
    %     \YFrench
    %     \young(12!\yb <10>!\yr <11>!\yw 2!\yb 4!\yr 6<12>!\yw <14>!\yb <11>!\yr
    %   <13>,!\yw 24!\yb 3!\yr 1!\yw 8!\yb <10>!\yr
    %   78!\yw 9!\yb 9,!\yw 67!\yb 4!\yr <10>!\yw 9!\yb 9!\yr <13><13>,!\yw
    %   11!\yb 1!\yr <11>!\yw 5!\yb 8!\yr 9,!\yw 99!\yb 4!\yr 9)
    %     \vspace{2mm} \\
    %     \si = \si^{[3]}
    % \end{array}
    %     \ \leftrightarrow\
    %     \begin{array}{c}
    %     	    \YFrench
    %         \young(122<11>,2489,679,115,99) \vspace{2mm} \\
    %     \si_1
    % \end{array}\ \oplus\ \ 
    % \begin{array}{c}
    %         \YFrench
    %         \Yfillcolour{cyan!70}
    %         \young(<10>4<14>,3<10>9,49,18,4) \vspace{2mm} \\
    %     \si_2
    % \end{array}\ \oplus\ \ 
    % \begin{array}{c}
    %         \YFrench
    %         \Yfillcolour{BrickRed}
    %         \young(<11>6<12><13>,178,<10><13><13>,<11>9,9) \vspace{2mm} \\
    %     \si_3
    % \end{array}
    % \vspace{-4mm}
    %     \]
    %     \caption{The diagram of an entry-wise sum of partitions.}
    %     \label{fig:LegArmSum}
    % \end{figure}


Let $\la^1,\dots,\la^r$ be partitions, and let $\pi \in \PPP([r])$ be
a set partition. For each $B \in \pi$, we are going to color the
columns of $\la^B$ by numbers $b \in B$ as follows: we observe
that the Young diagram $\la^B$ can be constructed 
    by sorting the columns of the diagrams $\la^{b_1}$, \ldots,
    $\la^{b_t}$ in decreasing order, where $B=\{b_1,\dots,b_t\}$ and
    $b_1<\cdots<b_t$. When several columns have the same length, we
    use the total order of $B$,
    that is we put first the columns of $\la^{b_1}$,
    then those of $\la^{b_2}$ and so on. We say that a
    column of $\la^B$ is \emph{colored by $b \in B$} if this column
    is identified with the column of $\la^b$ in the above
    construction. Similarly, we say that a box $\square \in \la^B$ is
    \emph{colored by $b \in B$} if it lies in the column colored
    by $b$; see \cref{subfig:A1}
    (at the moment, please disregard entries).
    This gives a way to identify boxes of $\la^{[r]}$ with the boxes of $\{\la^B:
    B \in \pi\}$. To be more precise a box $\square \in \la^B$ which
    lies in the $i$-th column colored by $b$ in $\la^B$ (not necessarily in the
    $i$-th column of $\la^B$) and in the $j$-th row of $\la^B$ is
    identified with the box $\tilde{\square}$ of $\la^{[r]}$ which
    lies in the $i$-th column colored by $b$ in $\la^{[r]}$ and in
    the $j$-th row of $\la^{[r]}$. 

This identification leads to a one-to-one correspondence between all the
fillings $\si$ of $\la^{[r]}$ with entries from a given set $\mathcal{A}$
and between the sets of fillings $\{\si_B: \la^B \to \A\ |\ B \in
\pi\}$, and for a given set of fillings $\{\si_b: \la^b \to
\mathcal{A}\ |\ b \in B\}$ the corresponding filling $\si : \la^B \to
\mathcal{A}$ is denoted by $\si^B$, see \cref{subfig:A1}.

\begin{figure}[t]
\subfloat[]{
\label{subfig:A1}
    \YFrench
    \Yboxdim{1cm}
    \begin{tikzpicture}[scale=0.5]
        \newcommand\yb{\Yfillcolour{cyan!70}}
        \newcommand\yr{\Yfillcolour{BrickRed}}
        \newcommand\yw{\Yfillcolour{white}}
        \newcommand\eleven{11}
        \tgyoung(0cm,0cm,12!\yb <10>!\yr \eleven!\yw 2!\yb 4!\yr 6<12>!\yw <14>!\yb \eleven!\yr
      <13>,!\yw 24!\yb 3!\yr 1!\yw 8!\yb <10>!\yr
      78!\yw 9!\yb 9,!\yw 67!\yb 4!\yr <10>!\yw 9!\yb 9!\yr <13><13>,!\yw
      11!\yb 1!\yr \eleven!\yw 4!\yb 4!\yr 4,!\yw 93!\yb 4!\yr 9)
\end{tikzpicture}}
\hspace{4pt}
\subfloat[]{
\label{subfig:A2}
    \YFrench
    \Yboxdim{1cm}
    \begin{tikzpicture}[scale=0.5, white/.style={circle,draw=black,fill=white,inner sep=1.5pt},
blue/.style={circle,draw=black,fill=cyan!70,inner sep=1.5pt}, red/.style={circle,draw=black,fill=BrickRed,inner sep=1.5pt}]
        \newcommand\yb{\Yfillcolour{cyan!70}}
        \newcommand\yr{\Yfillcolour{BrickRed}}
        \newcommand\yw{\Yfillcolour{white}}
            
        \tgyoung(0cm,0cm,;;;;;;;;;;;,;;;;;;;;;;,;;;;;;;;,;;;;;;;,;;;;)

        \draw[gray, thick] (0.5,4.5) to (1.5,4.5);
        \draw[gray, thick] (0.5,4.5) to [bend left] (2.5,4.5);
        \draw[gray, thick] (0.5,3.5) to [bend left] (3.5,3.5);
        \draw[gray, thick] (1.5,3.5) to [bend left] (3.5,3.5);
        \draw[gray, thick] (2.5,3.5) to (3.5,3.5);
        \draw[gray, thick] (0.5,3.5) to [bend left] (3.5,3.5);
        \draw[gray, thick] (0.5,2.5) to [bend left] (2.5,2.5);
        \draw[gray, thick] (3.5,2.5) to (4.5,2.5);
        \draw[gray, thick] (3.5,2.5) to [bend left] (5.5,2.5);
        \draw[gray, thick] (1.5,1.5) to (2.5,1.5);
        \draw[gray, thick] (2.5,1.5) to (3.5,1.5);
        \draw[gray, thick] (4.5,1.5) to [bend left]  (6.5,1.5);
        \draw[gray, thick] (5.5,1.5) to (6.5,1.5);
        \draw[gray, thick] (5.5,1.5) to [bend left] (7.5,1.5);
        \draw[gray, thick] (5.5,1.5) to [bend left] (8.5,1.5);
        \draw[gray, thick] (5.5,1.5) to [bend left] (9.5,1.5);
        \draw[gray, thick] (2.5,0.5) to [bend left] (4.5,0.5);
        \draw[gray, thick] (2.5,0.5) to [bend left] (5.5,0.5);
        \draw[gray, thick] (2.5,0.5) to [bend left] (6.5,0.5);
        \draw[gray, thick] (3.5,0.5) to (4.5,0.5);
        \draw[gray, thick] (3.5,0.5) to [bend left] (5.5,0.5);
        \draw[gray, thick] (3.5,0.5) to [bend left] (6.5,0.5);
        \draw[gray, thick] (7.5,0.5) to [bend left] (9.5,0.5);
        \draw[gray, thick] (8.5,0.5) to (9.5,0.5);
        \draw[gray, thick] (8.5,0.5) to [bend left] (10.5,0.5);

\draw (0.5,2.5) node [white] {};
\draw (0.5,3.5) node [white] {};
\draw (0.5,4.5) node [white] {};
\draw (1.5,1.5) node [white] {};
\draw (1.5,3.5) node [white] {};
\draw (1.5,4.5) node [white] {};
\draw (4.5,0.5) node [white] {};
\draw (4.5,1.5) node [white] {};
\draw (4.5,2.5) node [white] {};
\draw (8.5,0.5) node [white] {};
\draw (8.5,1.5) node [white] {};
\draw (2.5,0.5) node [blue] {};
\draw (2.5,1.5) node [blue] {};
\draw (2.5,2.5) node [blue] {};
\draw (2.5,3.5) node [blue] {};
\draw (2.5,4.5) node [blue] {};
\draw (5.5,0.5) node [blue] {};
\draw (5.5,1.5) node [blue] {};
\draw (5.5,2.5) node [blue] {};
\draw (9.5,0.5) node [blue] {};
\draw (9.5,1.5) node [blue] {};
\draw (3.5,0.5) node [red] {};
\draw (3.5,1.5) node [red] {};
\draw (3.5,2.5) node [red] {};
\draw (3.5,3.5) node [red] {};
\draw (6.5,0.5) node [red] {};
\draw (6.5,1.5) node [red] {};
\draw (7.5,0.5) node [red] {};
\draw (7.5,1.5) node [red] {};
\draw (10.5,0.5) node [red] {};
\end{tikzpicture}}
\hspace{-4pt}
\subfloat[]{
\label{subfig:A3}
    \begin{tikzpicture}[scale=0.3, white/.style={circle,draw=black,fill=white,inner sep=3pt},
blue/.style={circle,draw=black,fill=cyan!70,inner sep=3pt}, red/.style={circle,draw=black,fill=BrickRed,inner sep=3pt}]

\draw (3,0) node (W) [white] {};
\draw (0,5) node (B) [blue] {};
\draw (6,5) node (R) [red] {};

                \draw[gray, thick] (W) to [bend left=8] (B);
                \draw[gray, thick] (W) to [bend left=24] (B);
                \draw[gray, thick] (W) to [bend left=40] (B);
                \draw[gray, thick] (W) to [bend right=8] (B);
                \draw[gray, thick] (W) to [bend right=24] (B);
                \draw[gray, thick] (W) to [bend right=40] (B);
                \draw[gray, thick] (W) to [bend left=8] (R);
                \draw[gray, thick] (W) to [bend left=24] (R);
                \draw[gray, thick] (W) to [bend left=40] (R);
                \draw[gray, thick] (W) to [bend right=8] (R);
                \draw[gray, thick] (W) to [bend right=24] (R);
                \draw[gray, thick] (W) to [bend right=40] (R);
                \draw[gray, thick] (B) to [bend left=8] (R);
                \draw[gray, thick] (B) to [bend left=24] (R);
                \draw[gray, thick] (B) to [bend left=40] (R);
                \draw[gray, thick] (B) to [bend left=56] (R);
                \draw[gray, thick] (B) to [bend right=8] (R);
                \draw[gray, thick] (B) to [bend right=24] (R);
                \draw[gray, thick] (B) to [bend right=40] (R);
                \draw[gray, thick] (B) to [bend right=56] (R);
                \draw[gray, thick] (B) to
                [out=120,in=160,looseness=12] (B);
                \draw[gray, thick] (B) to
                [out=100,in=180,looseness=15] (B);
                \draw[gray, thick] (R) to
                [out=0,in=80,looseness=15] (R);
                \draw[gray, thick] (W) to [out=310,in=230,looseness=15] (W);



\end{tikzpicture}}
        \caption{\cref{subfig:A1} shows the colored filling $\si$ from
          \cref{fig:LegArmSum} with $\la^1 =
          (4^2,3^2,2), \la^2 = (3^2,2^2,1), \la^3 = (4,3^2,2,1)$. \cref{subfig:A2} shows the edges representing
          inversion pairs in $\si$ and colors of their 
          endpoints. \cref{subfig:A3} presents the multigraph $G^\si_{\la^1,\la^2,\la^3}$ for $\si$, and $\la^1,\la^2,\la^3$
          from \cref{subfig:A1}
          obtained from the edges from \cref{subfig:A2} by
          identifying the vertices of the same color.}
        \label{fig:Multigraph1}
    \end{figure}

Let $\sigma: \la^{[r]} \to \N_+$ be a filling. We are ready to
construct the
multigraph $G^\si_{\la^1,\dots,\la^r} := (V,E)$.  For each inversion pair in $\sigma$, we draw an edge
linking its boxes, and
we color its endpoints by the colors of these boxes
from the colored diagram $\la^{[r]}$; then we identify all the endpoints
of the same color -- see \cref{subfig:A2,subfig:A3} for a construction of
$G^\si_{\la^1,\dots,\la^r}$ for $r=3$ and $\si,\la^1,\la^2,\la^3$ as
in \cref{subfig:A1}.
More formally, $G^\si_{\la^1,\dots,\la^r} := (V,E)$ is defined by the
following data:
\begin{enumerate}
\item
\label{eq:graph1a}
the set of vertices $V$ is equal to $[r]$;
\item
\label{eq:graph1b}
the number $e_{i,j}(G)$ of edges connecting vertices $i$ and $j$ ($i$ and $j$ are not necessarily distinct)  is equal
to the number of inversion pairs in $\si$ colored by $\{i,j\}$.
\end{enumerate}

\subsection{Idea of the proof of \cref{theo:MacdonaldCumuFormula}}

The first important step towards the proof is the following lemma.
\begin{lemma}
\label{lem:TutteCumulant}
Let $G = (V,E)$ be a multigraph. Then
\begin{equation}
\label{eq:TutteCumulant}
\I_G(q) = (q-1)^{1-\#V} \sum_{\substack{\pi \in \PPP(V) } } 
            (-1)^{\#\pi-1}(\#\pi-1)!\ \prod_{B \in \pi} q^{\#E|_B},
\end{equation}
where $E|_B \subset E$ consists of the edges with
both endpoints lying in the set $B \subset V$.
\end{lemma}

\begin{proof}
Let $G$ be a multigraph (possibly disconnected), and we define the
generating functions $nc_G(q) = \sum_{H \subset G} q^{\#E(H)}$,
where we sum over all (possibly disconnected) sub-multigraphs of $G$ and 
$c_G(q) = \sum_{H \subset G} q^{\#E(H)}$,
where we sum over all connected sub-multigraphs of $G$.
Then, clearly
\[ nc_G(q) = \sum_{\pi \in \PPP(V)}\prod_{B \in \pi}c_{G|_B}(q),\]
where $G|_B$ is a sub-multigraph of $G$ supported on $B$.
Thus the M\"obius inversion formula on the set--partition lattice implies that
\[ c_G(q) = \sum_{\pi \in \PPP(V)}(-1)^{\#\pi-1}(\#\pi-1)!\ \prod_{B
    \in \pi}nc_{G|_B}(q).\]
Using \eqref{eq:DefTutte} we relate $c_G(q)$ with a Tutte polynomial:
\begin{equation} 
\label{eq:connect}
c_G(q) = (q)^{\#V-1}\Tu_G(1,q+1).
\end{equation} 
Plugging $nc_{G|_B}(q) = (1+q)^{\#E|_B}$ and \eqref{eq:connect} into
the above equality and substituting $\I_G(q)$ $= \Tu(1,q)$ yield the desired result.
\end{proof}

We are ready to sketch the proof of \cref{theo:MacdonaldCumuFormula}.

\begin{proof}[Sketch of the proof of \cref{theo:MacdonaldCumuFormula}]
We use definitions \eqref{eq:MacdoCumu} and \eqref{eq:DefCumulants} of Macdonald
cumulants and Haglund's formula \eqref{eq:Haglund} to rewrite the left hand side of \eqref{eq:MacdonaldCumuFormula}
as follows:
\begin{multline} 
\label{eq:takietakie}
(q-1)^{1-r}\sum_{\substack{\pi \in \PPP([r]) } } 
            (-1)^{\#\pi-1}(\#\pi-1)! \prod_{B \in \pi}
            \tilde{H}_{\la^B}(\bm{x};q,t) \\
= \sum_{\si:\la^{[r]}\to \N_+\ }t^{\maj(\si)}\left((q-1)^{1-r}\sum_{\substack{\pi \in \PPP([r]) } } 
            (-1)^{\#\pi-1}(\#\pi-1)! \prod_{B \in \pi}
            q^{\inv(\si^B)}\right)\xx^{\si}.
\end{multline}
This is a consequence of the one-to-one correspondence between
fillings of a given diagram and the sets of fillings of its
subdiagrams described in \cref{subsect:Identification} and the following identity:
\[ \maj(\si) = \sum_{\square \in
  \Des(\si)}(\ell_{\la^{[r]}}(\square)+1)
= \sum_{B \in \pi}\maj(\si^B).\]
Notice now that setting
$G^\si_{\la^1,\dots,\la^r} := (V,E)$, the expression in parentheses in \eqref{eq:takietakie} is given by the following formula:
\[ (q-1)^{1-\#V} \sum_{\substack{\pi \in \PPP(V) } } 
            (-1)^{\#\pi-1}(\#\pi-1)! \prod_{B \in \pi} q^{\#E|_B},\]
which is equal to
$\I_{G^\si_{\la^1,\dots,\la^r}}(1,q)$ by \cref{lem:TutteCumulant}. Indeed, strictly from the
construction of $G^\si_{\la^1,\dots,\la^r} $ one has $V = [r]$, and it
is easy to check that $\#E|_B = \inv(\si_B)$. This concludes the proof.
\end{proof}


\section{Consequences of the formula}
\label{sec:Fundamental}

In this section we present some important consequences of \cref{theo:MacdonaldCumuFormula}

\subsection{Expansion in fundamental quasisymmetric functions}

\begin{definition}[\cite{Gessel1984}] 
For any nonnegative integer $n$ and a subset $D \subset [n-1]$ a \emph{fundamental quasisymmetric
function} $F_{n,D}(\xx)$ of degree $n$ in variables $\xx = x_1,
x_2,\dots$ is defined by the formula
\begin{equation}
\label{eq:quasisymmetric}
F_{n,D}(\xx) := \sum_{\substack{i_1\leq \cdots \leq i_n\\j \in D \Rightarrow
    i_j < i_{j+1}}}x_{i_1}\cdots x_{i_n}.
\end{equation}
\end{definition}

Let $\la^1,\dots,\la^r$ be partitions and let $n= |\la^1|+\cdots+|\la^r|$. Then, using formula
\eqref{eq:MacdonaldCumuFormula} and a verbatim argumentation as in
\cite[Section 4]{HaglundHaimanLoehr2005} (see \cite{Dolega2017b} for
more details) we have the expansion:
\begin{equation}
\label{eq:CumulantInQuasisymmetric}
\ka(\la^1,\dots,\la^r)(\xx) = \sum_{\si \in \Sym{n}}\I_{G^{\si}_{\la^1,\dots,\la^r}}(q)\
t^{\maj_{\la^{[r]}}(\si)}\ F_{n,\iDes(\si)}(\xx),
\end{equation}
where we abuse notation by denoting both a permutation by $\si$, and
the associated standard filling of
$\la^{[r]}$ with the reading word given by $\si$ (see
\cref{subsec:Fillings}).

\subsection{Expansion in Schur symmetric functions}
\label{sec:Kostka}

    \begin{figure}[t]
\[
          \begin{array}{c}
        \YFrench
    \Yboxdim{1cm}
    \begin{tikzpicture}[scale=0.5]
        \newcommand\yb{\Yfillcolour{cyan!70}}
        \newcommand\yr{\Yfillcolour{BrickRed}}
        \newcommand\yw{\Yfillcolour{white}}
        \newcommand\eleven{11}
        \tgyoung(0cm,0cm,;;!\yb ;!\yr ;!\yw ;!\yb ;!\yr ;;!\yw ;!\yb ;!\yr
      ;,!\yw ;;!\yb ;!\yr ;!\yw ;!\yb ;!\yr
      ;;!\yw ;!\yb ;,!\yw ;;!\yb ;!\yr ;!\yw ;!\yb ;!\yr ;;,!\yw
      ;;!\yb ;!\yr ;!\yw ;!\yb ;!\yr ;,!\yw ;;!\yb ;!\yr ;)
       \Ylinethick{1.5pt}
       \tgyoung(1cm,1cm,!\yw ;)
       \tgyoung(5cm,2cm,!\yb ;)
       \tgyoung(3cm,3cm,!\yr ;)
\end{tikzpicture}
        \vspace{2mm}
    \end{array}
\
        \begin{array}{c}
        \YFrench
    \Yboxdim{1cm}
    \begin{tikzpicture}[scale=0.5, white/.style={circle,draw=black,fill=white,inner sep=1.5pt},
	blue/.style={circle,draw=black,fill=cyan!70,inner sep=1.5pt}, red/.style={circle,draw=black,fill=BrickRed,inner sep=1.5pt}]
        \newcommand\yb{\Yfillcolour{cyan!70}}
        \newcommand\yr{\Yfillcolour{BrickRed}}
        \newcommand\yw{\Yfillcolour{white}}
            
        \tgyoung(0cm,0cm,;;;;;;;;;;;,;;;;;;;;;;,;;;;;;;;,;;;;;;;,;;;;)

        \draw[gray, thick] (0.5,3.5) to [bend left] (3.5,3.5);
        \draw[gray, thick] (1.5,3.5) to [bend left] (3.5,3.5);
        \draw[gray, thick] (2.5,3.5) to (3.5,3.5);
        \draw[gray, thick] (0.5,3.5) to [bend left] (3.5,3.5);
        \draw[gray, thick] (0.5,2.5) to [bend left] (5.5,2.5);
        \draw[gray, thick] (1.5,2.5) to [bend left] (5.5,2.5);
        \draw[gray, thick] (2.5,2.5) to [bend left] (5.5,2.5);
        \draw[gray, thick] (3.5,2.5) to [bend left] (5.5,2.5);
        \draw[gray, thick] (4.5,2.5) to (5.5,2.5);
        \draw[gray, thick] (0.5,1.5) to (1.5,1.5);


		\draw (0.5,2.5) node [white] {};
		\draw (1.5,2.5) node [white] {};
		\draw (0.5,3.5) node [white] {};
		\draw (1.5,1.5) node [white] {};
		\draw (0.5,1.5) node [white] {};
		\draw (1.5,3.5) node [white] {};
		\draw (4.5,2.5) node [white] {};
		\draw (2.5,2.5) node [blue] {};
		\draw (2.5,3.5) node [blue] {};
		\draw (5.5,2.5) node [blue] {};
		\draw (3.5,2.5) node [red] {};
		\draw (3.5,3.5) node [red] {};
\end{tikzpicture} \vspace{2mm} \\
\
\end{array}
        \begin{array}{c}
    \begin{tikzpicture}[scale=0.35, white/.style={circle,draw=black,fill=white,inner sep=3pt},
	blue/.style={circle,draw=black,fill=cyan!70,inner sep=3pt}, red/.style={circle,draw=black,fill=BrickRed,inner sep=3pt}]

		\draw (3,1) node (W) [white] {};
		\draw (0,6) node (B) [blue] {};
		\draw (6,6) node (R) [red] {};

                \draw[gray, thick] (W) to (B);
                \draw[gray, thick] (W) to [bend left=16] (B);
                \draw[gray, thick] (W) to [bend right=16] (B);
                \draw[gray, thick] (W) to [bend left=8] (R);
                \draw[gray, thick] (W) to [bend right=8] (R);
                \draw[gray, thick] (B) to [bend left=8] (R);
                \draw[gray, thick] (B) to [bend right=8] (R);
                \draw[gray, thick] (B) to
                [out=100,in=180,looseness=15] (B);
                \draw[gray, thick] (W) to
                [out=310,in=230,looseness=15] (W);



\end{tikzpicture}
    \end{array}
    \vspace{-2mm}
        \]
        \caption{The graph
          $G_{\la^1,\la^2,\la^3}^{\square_1,\square_2,\square_3}$ for
          $\la^1,\la^2,\la^3$ from \cref{fig:Multigraph1}.}
        \label{fig:graph}
    \end{figure}

Let $\la^1,\dots,\la^r$ be partitions, and let $1 \leq s \leq
|\la^1|+\cdots+|\la^r|$ be a positive integer. For any
subset $\{\square_1,\dots,\square_s\} \subset \la^{[r]}$ of boxes we
construct a graph $G^{\square_1,\dots,\square_s}_{\la^1,\dots,\la^r} :=
(V,E)$ as follows: we draw an edge between each $\square_i$ and each
box to its left lying in the same row, and
we color its endpoints by the colors of the corresponding boxes
in $\la^{[r]}$; then we identify all the endpoints of the same
color -- see \cref{fig:graph} for a construction of
$G^{\square_1,\square_2,\square_3}_{\la^1,\dots,\la^r}$ for $r=3$ and $\la^1,\la^2,\la^3$ as in
\cref{fig:Multigraph1}. In other words
\begin{itemize}
\item
the set of vertices $V$ is equal to $[r]$, 
\item
the number of edges linking vertices $i,j \in V$ is equal to 
the number of pairs $(\square_k,\square')$ such that $\square'$ is
lying in the same row as $\square_k$ to its left, and the pair
$(\square_k,\square')$ is colored by $\{i,j\}$, where $1 \leq k \leq s$.
\end{itemize}

We are ready to prove \cref{conj:GeneralizedKostka} in the case of hooks.

\begin{theorem}
\label{theo:HookKostka}
Let $\la^1,\ldots,\la^r$ be partitions with
$|\la^{[r]}|=n$. Then, for any nonnegative integer $s$,
the coefficient of $(-u)^s$ in $\ka(\la^1,\dots,\la^r)[1-u]$ is equal
to
\begin{equation}
\label{eq:claim} 
\ka(\la^1,\dots,\la^r)[1-u]|_{(-u)^s} = \sum_{\{\square_1,\dots,\square_s\} \subset
  \la^{[r]}}\I_{G_{\la^1,\dots,\la^r}^{\square_1,\dots,\square_s}}(q)
\ t^{\sum_{1 \leq i
    \leq s}
    \ell'_{\la^{[r]}}(\square_i)}.
\end{equation}
Equivalently, the multivariate $q,t$-Kostka number
$\widetilde{K}_{(n-s,1^s); \la^1,\dots,\la^r}(q,t) := [s_{(n-s,1^s)}]\ka(\la^1,\dots,\la^r)$ is a polynomial in $q,t$ with
nonnegative integer coefficients given by the following formula:
\begin{equation}
\label{eq:HookKostka}
\widetilde{K}_{(n-s,1^s); \la^1,\dots,\la^r}(q,t)
=\sum_{\{\square_1,\dots,\square_s\} \subset
  \la^{[r]}\setminus(1,1)}\I_{G_{\la^1,\dots,\la^r}^{\square_1,\dots,\square_s}}(q)\
t^{\sum_{1 \leq i
    \leq s}
    \ell'_{\la^{[r]}}(\square_i)}.
\end{equation}
\end{theorem}

Let us describe the main ingredients of the proof of
\cref{theo:HookKostka}. For the detailed proof, we refer to \cite{Dolega2017b}.

\begin{proof}[Sketch of the proof of \cref{theo:HookKostka}]
Firstly, we sketch an argument of Macdonald showing the equivalence of
formulas \eqref{eq:claim} and \eqref{eq:HookKostka}.
Macdonald proved \cite[Section VI.8,
Example 2]{Macdonald1995} that
\[s_\la[1-u] = \begin{cases} 0 &\text{ if $\lambda$ is not a hook},\\
    (1-u)(-u)^s &\text{ if $\lambda = (n-s,1^s)$}.\end{cases}\]
Applying this to any homogeneous symmetric function
\[ f := \sum_{\lambda \vdash n} c_\la\ s_\la,\]
we obtain the following relation:
\begin{equation} 
\label{eq:relation}
f[1-u]|_{(-u)^s} = c_{(n-s,1^s)} + c_{(n-s+1,1^{s-1})}.
\end{equation}
Thus, we need to prove that the relation \eqref{eq:relation} is satisfied by
\eqref{eq:claim} and \eqref{eq:HookKostka}. This follows easily from
the construction of $G_{\la^1,\dots,\la^r}^{\square_1,\dots,\square_s}$.

Haglund, Haiman and Loehr \cite{HaglundHaimanLoehr2005} explained how to
compute $\tilde{H}_\mu(\xx;q,t)[1-u]|_{(-u)^s}$ from their formula
\eqref{eq:Haglund}. Using small modifications of their argumentation
and simple properties of graphs
$G_{\la^1,\dots,\la^r}^{\square_1,\dots,\square_s}$, we are able to compute $\ka(\la^1,\dots,\la^r)[1-u]|_{(-u)^s}$ relying
on \eqref{eq:MacdonaldCumuFormula} in a similar manner. This yields the
desired \eqref{eq:claim}.
\end{proof}

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