# LLT cumulants and graph coloring

Maciej Dołęga  Institute of Mathematics, Polish Academy of Sciences, ul.
Śniadeckich 8, 00-956 Warszawa, Poland<EMAIL_ADDRESS>and  Maciej Kowalski
Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8,
00-956 Warszawa, Poland<EMAIL_ADDRESS>
###### Abstract.

The purpose of this note is to introduce a new family of quasi-symmetric
functions called _LLT cumulants_ and discuss its properties. We define LLT
cumulants using the algebraic framework for conditional cumulants and we prove
that the Macdonald cumulant has an explicit positive expansion in terms of LLT
cumulants of ribbon shapes, generalizing the classical decomposition of
Macdonald polynomials. We also find a natural combinatorial interpretation of
the LLT cumulant of a given directed graph as a weighted generating function
of colorings of its subgraphs.

We use this graph theoretical framework to prove various positivity results.
This includes monomial positivity, positivity in fundamental quasisymmetric
functions and related positivity of the coefficients of Schur polynomials
indexed by hook shapes. We also prove $e$-positivity for vertical-shape LLT
cumulants, after the shift of variable $q\to q+1$, which refines a recent
result of Alexandersson and Sulzgruber. All these results give evidence
towards Schur-positivity of LLT cumulants, which we conjecture here. We prove
that this conjecture implies Schur-positivity of Macdonald cumulants, and we
give more evidence by proving the conjecture for LLT cumulants of melting
lollipops that refines a recent result of Huh, Nam and Yoo.

The authors are supported by Narodowe Centrum Nauki, grant
UMO-2017/26/D/ST1/00186.

smalltableaux

##  1\. Introduction

In 1988, Macdonald [Mac88] introduced his celebrated two-parameter symmetric
functions and conjectured that when expanded in the basis of Schur symmetric
functions, their coefficients have a remarkable property: they seem to be
polynomials in two deformation parameters $q,t$ with nonnegative integer
coefficients. Since then, a broad community working on the symmetric functions
theory devoted themselves to prove Macdonald’s conjecture, which resulted in a
huge development of the field.

In 1995, Lapointe and Vinet [LV95] proved that the coefficients of Jack
symmetric functions expanded in the monomial basis are polynomials in the
deformation parameter $\alpha$ with integer coefficient. Two years later, Knop
and Sahi [KS97] found an explicit positive formula for this expansion. Since
Jack symmetric functions are a limit case of Macdonald symmetric functions,
these results inspired further research and shortly afterwards, the
polynomiality of the coefficients of Macdonald polynomials was proved
independently and almost simultaneously (using different approaches) in five
different papers [Sah96, GT96, LV97, Kno97, KN98]. An affirmative answer to
Macdonald’s original conjecture was finally released in a beautiful and
difficult paper of Haiman [Hai01], who was able to relate Macdonald’s question
to a question about the geometry of Hilbert schemes of points in the complex
plane, to which he gave an affirmative answer. Even though this result built
new bridges between various fields of mathematics, it did not provide an
explicit combinatorial formula explaining Schur-positivity. Regardless, it
generated new research directions related with the structure of Macdonald and
related symmetric functions.

In 2005, Haglund, Haiman and Loehr [HHL05a] found an explicit combinatorial
formula for Macdonald polynomials, lifting Knop and Sahi’s formula to the two-
parameter world of Macdonald, and relating Macdonald polynomials with another
family of symmetric functions introduced by Leclerc, Lascoux and Thibon in
1997 [LLT97], and later conveniently named _LLT polynomials_. Haglund, Haiman
and Loehr noticed [HHL05a] that Macdonald polynomials can be naturally
decomposed as a positive combination of LLT polynomials, so proving Schur-
positivity for LLT polynomials would give yet another proof of the famous
conjecture of Macdonald. This was done by Grojnowski and Haiman [GH07], who
related LLT polynomials with the Kazhdan-Lusztig theory in a much more general
setting than what was done before [LT00], and therefore proved the Schur-
positivity of LLT polynomials indexed by arbitrary skew-shapes (see Section
2.1 for the details and all the necessary definitions).

In 2017, the first author together with Féray [DF17] introduced _Jack
cumulants_ as a tool to approach a fascinating open problem in the theory of
symmetric functions known as _the $b$-conjecture_ (posed by Goulden and
Jackson [GJ96]), which relates Jack symmetric functions with a weighted
generting function of graphs embedded into surfaces (and which, despite some
recent progress [CD22], is still wide open). The notion of Jack cumulants
naturally extends to _Macdonald cumulants_ the same way as Jack polynomials
can be seen as the limit case of Macdonald polynomials. The first author with
Féray observed conjecturally that the coefficients of Macdonald cumulants seem
to be polynomials, which was later proved in [Doł17] and further improved in
[Doł19], where an explicit positive combinatorial formula for the Macdonald
cumulants was proved. This rich combinatorial structure of Macdonald cumulants
naturally calls for investigating the expansion in the Schur basis: extensive
computer simulations performed by the first author [Doł17] have led him to
believe that a more general version of the original question of Macdonald is
true: the coefficients of the Schur expansion of Macdonald cumulants are
polynomials in $q,t$ with nonnegative integer coefficients. We were recently
informed that the logarithm of a partition function for Macdonald polynomials
was considered by Hausel, Letellier and Rodriguez Villegas [HLRV11], who
conjectured its monomial positivity interpreted as the mixed Hodge polynomials
of character varieties. The notion of Macdonald cumulants appears naturally in
the decomposition of the logarithm of the partition function, and the recent
work [AMRV19] (following [HS02, CBVdB04]) exhibits that the Poincaré
polynomial of Nakajima quiver variaties (which can be seen as a special case
of the aforementioned conjecture) is given by the specialization of the Tutte
polynomial $\operatorname{Tutte}_{G}(1,q)$, which is the same phenomenon as in
our combinatorial interpretation of Macdonald cumulants [Doł19]. All this
gives yet additional motivation for studying the combinatorial structure of
Macdonald cumulants.

The main purpose of this note is to take a step in this direction by
introducing the notion of LLT cumulants. There are two natural motivations for
introducing them:

  * •

by analogy with the decomposition of Macdonald polynomials into LLT
polynomials, we show that the same phenomenon occurs at the level of higher
cumulants: _Macdonald cumulants_ can be naturally expressed as a positive
linear combination of _LLT cumulants_ – see Theorem 2.9;

  * •

in contrast to the purely algebraic definition of Macdonald cumulants inspired
by the theory of conditional cumulants, we show that LLT cumulants (a priori
defined using the same abstract framework) can be equivalently defined purely
combinatorially as graph colorings – see Theorem 3.8. In particular, it is
natural to study a general class of graph colorings which contains LLT
polynomials and LLT cumulants, and that allows to treat certain LLT-specific
phenomena in a more general graph-theoretical sense – see Section 3.

There are several applications of the aforementioned results. We start by
developing the theory of $q$-partial cumulants, which generalize the
$G$-inversion polynomials or, equivalently, the generating series of
$G$-parking functions, which is also equal to the evaluation of the Tutte
polynomial $\operatorname{Tutte}_{G}(1,q)$ (see Section 2.2), and we prove a
positivity result for these cumulants (see Theorem 2.5). This result is
crucial for proving Theorem 2.10 which says that Schur positivity of the
cospin LLT cumulant (that we state as 2.7) implies Schur positivity of
Macdonald cumulants conjectured in [Doł17]. In Section 3, we introduce certain
digraphs that we call _LLT graphs_ , and we show that every LLT polynomial is
a weighted generating function of LLT graph colorings. We describe the ring
generated by these LLT graphs and we prove that the LLT cumulant of an
$r$-colored LLT graph $(G,f)$ has a natural interpretation as a weighted
generating function of colorings of all _$f$ -connected_ subgraphs of $G$ (see
Definition 3.7 and the preceding paragraph for the precise definition of
$f$-connectedness and LLT cumulants of $r$-colored LLT graphs). We obtain this
interpretation by studying certain relations between colorings of various LLT
graphs.

It is worth mentioning that recently, various authors have already proven many
interesting results concerning positivity of LLT polynomials and they heavily
relied on some relations between them [Lee21, HNY20, AN21, AS22, Tom21]. Our
interpretation of LLT polynomials and LLT cumulants proves that the graph-
theoretical point of view is very natural and a characterization of all
possible relations might potentially be achieved pushing these studies further
in the future (see Remark 3.3). We use our framework to refine some of the
previous positivity results, which gives evidence towards 2.7:

  * •

we prove that the coefficients of LLT cumulants of an $r$-colored LLT graph
$(G,f)$ in the quasi-symmetric monomial basis are polynomials in $q$ with
nonnegative integer coefficients and we provide their explicit combinatorial
interpretation (see Theorem 3.10). This result is a refinement of the
combinatorial formula for Macdonald polynomials [Doł19];

  * •

we deduce an analogous result for the fundamental quasisymmetric basis and
using standard procedures, we deduce positivity of the coefficients of Schur
basis indexed by hooks (see Theorems 3.13 and 3.15);

  * •

we prove that LLT polynomials considered after the shift $q\to q+1$ naturally
decompose as a sum of products of LLT cumulants. In the special case of
vertical-strips, we deduce from the recent result of Alexandersson and
Sulzgruber [AS22] a positive combinatorial formula for LLT cumulants in the
basis of elementary functions (see Theorem 3.17);

  * •

we prove Schur positivity of LLT cumulants of $r$-colored lollipop graphs,
generalizing previous result of Huh, Nam and Yoo [HNY20] (see Section 4.1 for
the definitions and Theorem 4.7 for the result).

Our paper is organized as follows: in Section 2, we review the necessary
background on Macdonald and LLT polynomials and on cumulants. Then we
introduce $q$-partial cumulants, we state our main 2.7, and we prove that it
implies Schur positivity of Macdonald cumulants. Section 3 is devoted to the
study of LLT graphs and weighted generating functions of their colorings that
we introduce. In Section 3.1, we give a combinatorial interpretation of LLT
cumulants in the graph-theoretical framework and in Section 3.2, we prove
various positivity results supporting 2.7. In Section 4, we conclude with
comments and questions related with 2.7 and, in particular, further partial
results including Schur positivity for LLT cumulants of $r$-colored melting
lollipops.

##  2\. Macdonald cumulants and expansion in LLT cumulants

We use French convention for drawing Young diagrams, i.e. the largest row is
at the bottom and the largest column is on the left hand side.

###  2.1. LLT and Macdonald polynomials

Let $\bm{\nu}=(\lambda^{1}/\mu^{1},\dots,\lambda^{\ell}/\mu^{\ell})$ be an
$\ell$-tuple of skew Young diagrams (and denote $\ell(\bm{\nu}):=\ell$). For
each box $\square=(x,y)\in\lambda^{i}/\mu^{i}$, we define its content
$c(\square)=x-y$ and its shifted content as $\tilde{c}(\square)=\ell
c(\square)+i-1$. We say that a box $\square\in\bm{\nu}$ _attacks_ a box
$\square^{\prime}\in\bm{\nu}$ if
$0<\tilde{c}(\square^{\prime})-\tilde{c}(\square)<\ell$. Let $T$ be a filling
of cells of the diagrams in $\bm{\nu}$. If for each $i\in[1..\ell]$ the
entries in $\lambda^{i}/\mu^{i}$ are weakly increasing in rows (from left to
right) and strictly increasing in columns (from bottom to top), we say that
$T$ is a _semistandard_ filling, and we denote it by
$T\in\operatorname{SSYT}(\bm{\nu})$. Finally, we call a pair of boxes
$\square,\square^{\prime}\in\bm{\nu}$ an _inversion_ of $T$ if
$T(\square)>T(\square^{\prime})$ and $\square$ attacks $\square^{\prime}$. We
denote the set of inversions of $T$ by $\operatorname{Inv}(T)$ and its
cardinality by $\operatorname{inv}(T)$.

_LLT polynomial_ $\operatorname{LLT}(\bm{\nu})$ is the weighted generating
series of $\operatorname{SSYT}(\bm{\nu})$:

(1)
$\operatorname{LLT}(\bm{\nu})=\sum_{T\in\operatorname{SSYT}(\bm{\nu})}q^{\operatorname{inv}(T)}\bm{x}^{T},$

where $\bm{x}^{T}:=\prod_{\square\in\bm{\nu}}x_{T(\square)}$.

This definition was introduced in [HHL05a] and it is related to the original
definition of Lascoux, Leclerc and Thibon [LLT97, Equation (26)] by:

(2)
$\operatorname{LLT}^{\operatorname{cospin}}(\bm{\nu})=q^{-\min_{T\in\operatorname{SSYT}(\bm{\nu})}\operatorname{inv}(T)}\sum_{T\in\operatorname{SSYT}(\bm{\nu})}q^{\operatorname{inv}(T)}\bm{x}^{T},$

where
$\operatorname{LLT}^{\operatorname{cospin}}(\bm{\nu})=\tilde{G}^{(r)}_{\rho}(X;q)$
using notation from [LLT97, Equation (26)].

######  Remark 2.1.

The shape $\rho$ is obtained from $\bm{\nu}$ via the Stanton–White algorithm
[SW85], and since we do not use the original version of $r$-ribbon tableaux in
this article, we treat Eq. 1 as the definition and refer to [SW85, LLT97] for
those who are interested in the equivalent framework of $r$-ribbon tableaux.

The statistic
$\operatorname{inv}(T)-\min_{T\in\operatorname{SSYT}(\bm{\nu})}\operatorname{inv}(T)$
can be realized as the cardinality of a subset
$\operatorname{Inv}_{\operatorname{cospin}}(T)$ of $\operatorname{Inv}(T)$ due
to [SSW03] (in particular,
$\min_{T\in\operatorname{SSYT}(\bm{\nu})}\operatorname{inv}(T)=|\operatorname{Inv}(T)\setminus\operatorname{Inv}_{\operatorname{cospin}}(T)|$
for any $T\in\operatorname{SSYT}(\bm{\nu})$). For a box $\square\in\bm{\nu}$,
we denote by
$\square_{\leftarrow},\square_{\rightarrow},\square_{\uparrow},\square_{\downarrow}$
the boxes which are lying directly to the left, right, up and down of the box
$\square$, respectively. Define
$\operatorname{Inv}_{\operatorname{cospin}}(T)$ as follows:

$\displaystyle\operatorname{Inv}_{\operatorname{cospin}}(T)=\\{$
$\displaystyle(\square,\square^{\prime})\in\operatorname{Inv}(T):(\square^{\prime}_{\uparrow},\square)\in\operatorname{Inv}(T)\text{
and the row coordinate}$ $\displaystyle\text{of $\square$ is weakly smaller
than the row coordinate of $\square^{\prime}$}\\}.$

Here, the convention is that for $\square^{\prime}_{\uparrow}\notin\bm{\nu}$
the pair $(\square^{\prime}_{\uparrow},\square)$ is automatically an
inversion. Then

(3)
$\operatorname{LLT}^{\operatorname{cospin}}(\bm{\nu})=\sum_{T\in\operatorname{SSYT}(\bm{\nu})}q^{|\operatorname{Inv}_{\operatorname{cospin}}(T)|}\bm{x}^{T}.$

In the special case when
$\bm{\nu}=(\lambda^{1}/\mu^{1},\dots,\lambda^{\ell}/\mu^{\ell})$ is a sequence
of ribbon shapes, i.e., connected skew shapes which do not contain a shape of
size $2\times 2$, we define a normalization

(4)
$\operatorname{LLT}^{\operatorname{Mac}}(\bm{\nu})=q^{-a(\bm{\nu})}\sum_{T\in\operatorname{SSYT}(\bm{\nu})}q^{\operatorname{inv}(T)}\bm{x}^{T},$

where

$a(\bm{\nu})=\sum_{\square\in\operatorname{Des}(\bm{\nu})}|\\{\square^{\prime}:c(\square^{\prime})=c(\square),\tilde{c}(\square^{\prime})>\tilde{c}(\square)\\}|,$

and an element of $\operatorname{Des}(\bm{\nu})$ is a box in $\bm{\nu}$, which
is lying directly above another box in $\bm{\nu}$.

This particular choice of normalization is motivated by the combinatorial
formula of Haglund, Haiman and Loehr, for Macdonald polynomials
$\tilde{H}_{\lambda}^{(q,t)}$. It expresses a Macdonald polynomial
$\tilde{H}_{\lambda}^{(q,t)}$ as a sum of LLT polynomials indexed by
$\lambda_{1}$-tuples of shapes of sizes $\lambda^{\prime}_{j}$, $1\leq
j\leq\lambda_{1}$ where $\lambda^{\prime}$ denotes the _transpose_ of
$\lambda$, i.e., the diagram with $\lambda_{1}$ boxes in the first column,
$\lambda_{2}$ boxes in the second column, etc. For our purposes, we treat the
following formula as the definition of Macdonald polynomials:

######  Theorem 2.2.

[HHL05a] For any partition $\lambda$ the following expansion holds true

(5)
$\tilde{H}_{\lambda}^{(q,t)}=\sum_{\bm{\nu}}t^{\operatorname{maj}(\bm{\nu})}\operatorname{LLT}^{\operatorname{Mac}}_{\bm{\nu}},$

where we sum over all tuples of skew-partitions such that $\bm{\nu}_{j}$ is a
ribbon of length $\lambda^{\prime}_{j}$ whose bottom, far-right cell has
content $0$.

The statistic $\operatorname{maj}$, which appears in (5), is defined as
follows:

(6)
$\operatorname{maj}(\bm{\nu}):=\sum_{i=1}^{\ell(\bm{\nu})}\operatorname{maj}(\bm{\nu}_{i})=\sum_{i=1}^{\ell(\bm{\nu})}\sum_{\square\in\operatorname{Des}(\bm{\nu}_{i})}|\\{\square^{\prime}\in\bm{\nu}_{i}:c(\square^{\prime})<c(\square)|.$

###  2.2. Cumulants

The notion of cumulants was originally studied by Leonov and Shiryaev [LS59]
in the context of probability theory. Cumulants appear now in a wide variety
of contexts, see [JLuR00, Chapter 6] for their role in studying random graphs
and [NŚ11] for a concise introduction to noncommutative probability theory and
various types of cumulants. In what follows, we will be interested in the
$q$-deformation of partial cumulants that appeared in [Doł17] and was inspired
by the classical definition of conditional cumulants (see Definition 2.4).

######  Definition 2.3.

Suppose that $\mathcal{A}$ is an algebra over the fraction field
$\mathbb{Q}(q)$. Let $\bm{u}:=(u_{I})_{I\subseteq V}$ be a family of elements
in $\mathcal{A}$, indexed by subsets of a finite set $V$. Then its $q$-partial
cumulants are defined as follows. For any non-empty subset $I$ of $V$, set

(7)
$\kappa^{(q)}_{I}(\bm{u})=(q-1)^{1-|I|}\sum_{\begin{subarray}{c}\pi\in\mathcal{P}(I)\end{subarray}}(-1)^{|\pi|-1}(|\pi|-1)!\prod_{B\in\pi}u_{B}.$

The sum runs over elements of the family $\mathcal{P}(I)$ of _set-partitions_
of $I$: a set-partition $\pi\in\mathcal{P}(I)$ is a set of disjoint subsets of
$I$ whose union is equal to $I$ (so one can think that an element
$\pi\in\mathcal{P}(I)$ is grouping elements of $I$ into disjoint subsets) and
the number of elements of $\pi$ is denoted by $|\pi|$.

######  Definition 2.4.

Let $\mathcal{A}$ be a vector space with two different commutative
multiplicative structures $\cdot$ and $\oplus$, which define two (different)
algebra structures on $\mathcal{A}$. For any
$X_{1},\dots,X_{r}\in\mathcal{A}$, we define the _conditional cumulant_
$\kappa(X_{1},\dots,X_{r})\in\mathcal{A}$ as the coefficient of $t_{1}\cdots
t_{r}$ in the following formal power series in $t_{1},\dots,t_{r}$:

(8) $\kappa(X_{1},\dots,X_{r}):=[t_{1}\cdots
t_{r}]\log_{\cdot}\left(\exp_{\oplus}(t_{1}X_{1}+\cdots+t_{r}X_{r})\right),$

where $\log_{\cdot}$ and $\exp_{\oplus}$ are defined in a standard way with
respect to multiplication given by $\cdot$ and $\oplus$, respectively.

With the above in mind, we get

$\log_{\cdot}(1+A)=\sum_{n\geq 1}\frac{(-1)^{n-1}A^{\cdot
n}}{n},\qquad\exp_{\oplus}(A)=\sum_{n\geq 0}\frac{A^{\oplus n}}{n!}.$

Then, one can show that setting

$u_{B}:=\bigoplus_{b\in B}X_{b},$

the $q$-partial cumulant $\kappa^{(q)}_{[1..r]}(\bm{u})$ evaluated at $q=0$
coincides with the conditional cumulant $\kappa(X_{1},\dots,X_{r})$ up to a
sign:

$\kappa^{(0)}_{[1..r]}(\bm{u})=(-1)^{r-1}\kappa(X_{1},\dots,X_{r}).$

Although the cumulants originate from the probability theory, the
$q$-deformation introduced here is also relevant in the context of certain
graph invariants, called _inversion polynomials_. Let $G=(V,E)$ be a
multigraph (i.e. a graph with multiple loops and multiple edges allowed) and
for any subset of vertices $I\subset V$ we denote by $e_{I}$ the number of
edges in $G$ connecting vertices in $I$. It was shown in [Doł19] that for the
family $\bm{u}$ defined by

$u_{I}:=q^{e_{I}}$

the asociated $q$-partial cumulant $\kappa^{q}_{V}(\bm{u})$ is equal to the
$G$-inversion polynomial $\mathcal{I}_{G}(q)$ (which is also equal to the
evaluation of the Tutte polynomial $\operatorname{Tutte}_{G}(1,q)$ and to the
generating series of $G$-parking function; a fact that will not be used in
this paper). In particular, it is a polynomial in $q$ with nonnegative integer
coefficients and it was used to prove positivity results for the $q$-partial
cumulants of Macdonald polynomials; we postpone its precise definition to
Section 3.2.1, where we use it to provide certain explicit combinatorial
formulae. In the following, we show another positivity property of cumulants
constructed by using multigraphs. This positivity property will be crucial for
our first applications.

Suppose that $\bm{u}$ is a family as in Definition 2.3, and let $G$ be a
multigraph with the vertex set $V$. Define the family $\bm{u}^{G}$ by setting

$u_{I}^{G}:=q^{e_{I}}u_{I}$

for any subset $I\subset V$. Finally, for any set-partition
$\pi\in\mathcal{P}(I)$, define a family
$\bm{u}(\pi):=(\tilde{u}_{B})_{B\subset\pi}$ by setting
$\tilde{u}_{B}:=u_{\cup B}$ (note that for $B\subset\pi\in\mathcal{P}(I)$, one
has $\bigcup B\subset I$ so that $\bm{u}(\pi)$ is well defined).

######  Theorem 2.5.

The $q$-partial cumulant $\kappa^{(q)}_{I}(\bm{u}^{G})$ is a $q$-positive
combination of the $q$-partial cumulants $\kappa^{(q)}_{\pi}(\bm{u}(\pi))$,
where $\pi\in\mathcal{P}(I)$.

###### Proof.

We will prove the theorem by induction on $|I|$. For $|I|=1$, the statement is
obvious so suppose that $|I|>1$. Strictly from the definition of the
$q$-partial cumulant (7), $\kappa^{(q)}_{I}(\bm{u}^{G})$ can be expressed as
$\kappa^{(q)}_{I}(\bm{u}^{G})=q^{\sum_{i\in
I}e_{\\{i\\}}}\cdot\kappa^{(q)}_{I}(\bm{u}^{G^{\prime}})$, where $G^{\prime}$
is the graph $G$ restricted to the vertices from $I$ and with all the loops
removed. Indeed, for every set-partition $\pi\in\mathcal{P}(I)$, the summand
$(-1)^{|\pi|-1}(|\pi|-1)!\prod_{B\in\pi}q^{e_{B}}u_{B}$ appearing in (7) can
be rewritten as $q^{\sum_{i\in
I}e_{\\{i\\}}}(-1)^{|\pi|-1}(|\pi|-1)!\prod_{B\in\pi}q^{e_{B}(G^{\prime})}u_{B},$
where $e_{B}(G^{\prime})$ denotes the number of edges in $G^{\prime}$
connecting vertices in $B$ (which is the same as the number of edges in $G$
connecting distinct vertices in $B$). We further decompose
$\kappa_{I}^{(q)}(\bm{u}^{G})$ as

$\kappa_{I}^{(q)}(\bm{u}^{G})=q^{\sum_{i\in
I}e_{i}}\bigg{(}\kappa^{(q)}_{I}(\bm{u})+\Big{(}\kappa^{(q)}_{I}(\bm{u}^{G^{\prime}})-\kappa^{(q)}_{I}(\bm{u})\Big{)}\bigg{)},$

which is relevant for using the inductive hypothesis. Indeed, the second term
in this decomposition can be expressed as:

$\displaystyle\kappa^{(q)}_{I}(\bm{u}^{G^{\prime}})-\kappa^{(q)}_{I}(\bm{u})$
$\displaystyle=(q-1)^{1-|I|}\sum_{\begin{subarray}{c}\pi\in\mathcal{P}(I)\end{subarray}}(-1)^{|\pi|-1}(|\pi|-1)!\big{(}q^{\sum_{B\in\pi}e_{B}(G^{\prime})}-1\big{)}\prod_{B\in\pi}u_{B}$
$\displaystyle=(q-1)^{2-|I|}\sum_{\begin{subarray}{c}\pi\in\mathcal{P}(I),\\\
|\pi|<|I|\end{subarray}}(-1)^{|\pi|-1}(|\pi|-1)!\left[\sum_{B\in\pi}e_{B}(G^{\prime})\right]_{q}\prod_{B\in\pi}u_{B},$

where $[n]_{q}:=\frac{q^{n}-1}{q-1}=\sum_{i=0}^{n-1}q^{i}$ is the standard
numerical factor. Let $e(G^{\prime}):=e_{I}(G^{\prime})$ denote the number of
edges in $G^{\prime}$. In the following, we are going to construct set-
partitions $\sigma_{1},\dots,\sigma_{e(G^{\prime})}\in\mathcal{P}(I)$ each
consisting of precisely $|I|-1$ elements, and graphs
$G_{1},\dots,G_{e(G^{\prime})}$ with $|I|-1$ vertices such that

(9)
$\displaystyle(q-1)^{2-|I|}\sum_{\begin{subarray}{c}\pi\in\mathcal{P}(I),\\\
|\pi|<|I|\end{subarray}}(-1)^{|\pi|-1}(|\pi|-1)!\left[\sum_{B\in\pi}e_{B}(G^{\prime})\right]_{q}\prod_{B\in\pi}u_{B}=\sum_{k=1}^{e(G^{\prime})}\kappa^{(q)}_{\sigma_{k}}(\bm{u}(\sigma_{k})^{G_{k}}),$

which allows to conclude the proof using the inductive hypothesis.

We arbitrarily order edges of $G^{\prime}$ and for any $1\leq i\leq
e(G^{\prime})$ we denote by $E_{i}(G^{\prime})$ the set of the first $i$ edges
in $G^{\prime}$. Let $\\{m,n\\}$ be the set of endpoints of the $i$-th edge in
$G^{\prime}$. Define the graph $G_{i}$ as follows:

  * •

its set of vertices is equal to the set partition
$\sigma_{i}:=\\{\\{m,n\\},\\{k\\}\colon k\in I\setminus\\{m,n\\}\\}$, which is
the unique set partition of $I$ with $|I|-1$ elements, whose element of size
two is equal to $\\{m,n\\}$. In other terms, there is precisely one vertex of
$G_{i}$ equal to the set $\\{m,n\\}$, and every other vertex of $G_{i}$ is
equal to the singleton $\\{k\\}$, where $k\in I\setminus\\{m,n\\}$. In
particular, $G_{i}$ has precisely $|I|-1$ vertices.

  * •

For any elements $k,l\in I\setminus\\{m,n\\}$, the number of edges linking
vertices $\\{k\\}$ and $\\{l\\}$ in $G_{i}$ is given by the number of edges in
$E_{i}(G^{\prime})$ with endpoints $\\{k,l\\}$.

  * •

For each $k\in I\setminus\\{m,n\\}$, the number of edges linking vertices
$\\{k\\}$ and $\\{m,n\\}$ in $G_{i}$ is given by the number of edges in
$E_{i}(G^{\prime})$ with endpoints $\\{k,m\\}$ or $\\{k,n\\}$.

  * •

Finally, the number of loops attached to vertex $\\{m,n\\}$ is given by the
number of edges in $E_{i-1}(G^{\prime})$ with endpoints $\\{m,n\\}$.

Let us prove by induction on $e(G^{\prime})$ that the constructed graphs
satisfy Eq. 9. Clearly, when $G^{\prime}$ has no edges, both hand sides of Eq.
9 are equal to $0$. Suppose that $e(G^{\prime})>0$ and let $G^{\prime\prime}$
denote the graph obtained from $G^{\prime}$ by removing its largest edge
$\\{m,n\\}$. Then

$\displaystyle(q-1)^{2-|I|}\sum_{\begin{subarray}{c}\pi\in\mathcal{P}(I),\\\
|\pi|<|I|\end{subarray}}(-1)^{|\pi|-1}(|\pi|-1)!\left[\sum_{B\in\pi}e_{B}(G^{\prime})\right]_{q}\prod_{B\in\pi}u_{B}=$
$\displaystyle(q-1)^{2-|I|}\sum_{\begin{subarray}{c}\pi\in\mathcal{P}(I),\\\
|\pi|<|I|\end{subarray}}(-1)^{|\pi|-1}(|\pi|-1)!\left[\sum_{B\in\pi}e_{B}(G^{\prime\prime})\right]_{q}\prod_{B\in\pi}u_{B}+$
$\displaystyle(q-1)^{2-|I|}\sum_{\pi\in\mathcal{P}(\sigma_{e(G^{\prime})})}(-1)^{|\pi|-1}(|\pi|-1)!\prod_{B\in\pi}q^{e_{\cup
B}(G^{\prime\prime})}u_{\cup B}.$

By the inductive hypothesis, we have that

$\displaystyle(q-1)^{2-|I|}\sum_{\begin{subarray}{c}\pi\in\mathcal{P}(I),\\\
|\pi|<|I|\end{subarray}}(-1)^{|\pi|-1}(|\pi|-1)!\left[\sum_{B\in\pi}e_{B}(G^{\prime\prime})\right]_{q}\prod_{B\in\pi}u_{B}=\sum_{k=1}^{e(G^{\prime\prime})}\kappa^{(q)}_{\sigma_{k}}(\bm{u}(\sigma_{k})^{G_{k}}).$

Moreover, strictly from the construction of $G_{e(G^{\prime})}$, we have that
$e_{B}(G_{e(G^{\prime})})=e_{\cup B}(G^{\prime})-1=e_{\cup
B}(G^{\prime\prime})$ for any $\\{\\{m,n\\}\\}\subset
B\subset\sigma_{e(G^{\prime})}$ and $e_{B}(G_{e(G^{\prime})})=e_{\cup
B}(G^{\prime})=e_{\cup B}(G^{\prime\prime})$ for any
$B\subset\sigma_{e(G^{\prime})}\setminus\\{\\{m,n\\}\\}$. Therefore,

$\displaystyle(q-1)^{2-|I|}\sum_{\pi\in\mathcal{P}(\sigma_{e(G^{\prime})})}(-1)^{|\pi|-1}(|\pi|-1)!\prod_{B\in\pi}q^{e_{\cup
B}(G^{\prime\prime})}u_{\cup
B}=\kappa^{(q)}_{\sigma_{e(G^{\prime})}}(\bm{u}(\sigma_{e(G^{\prime})})^{G_{e(G^{\prime})}}),$

which finishes the proof. ∎

###  2.3. Macdonald and LLT cumulants

Let $\bm{\nu}=(\lambda^{1}/\mu^{1},\dots,\lambda^{\ell}/\mu^{\ell})$ be an
$\ell$-tuple of skew Young diagrams. For any surjective function
$f:[1..\ell]\to[1..r]$, we say that a pair $(\bm{\nu},f)$ is an _$r$ -colored
tuple_ of skew Young diagrams and we will think of it as an $\ell$-tuple
colored by $r$ colors, so that $i$-th element $\lambda^{i}/\mu^{i}$ has color
$f(i)$. For an $r$-colored tuple of skew Young diagrams $(\bm{\nu},f)$ and for
a subset $B\subset[1..r]$, we define a tuple of skew Young diagrams
$(\bm{\nu},f)^{B}$ as the sub-tuple of $\bm{\nu}$ colored by colors from $B$.
More formally,
$(\bm{\nu},f)^{B}:=(\lambda^{i_{1}}/\mu^{i_{1}},\dots,\lambda^{i_{k}}/\mu^{i_{k}})$,
where $f^{-1}(B)=\\{i_{1},\dots,i_{k}\\}$ and $i_{1}<\cdots<i_{k}$.

For a given $r$-colored tuple of skew Young diagrams $(\bm{\nu},f)$, we define
_LLT cumulants_ (with respect to different normalizations) by the following
formulae:

(10)
$\displaystyle\kappa_{\operatorname{LLT}^{\operatorname{cospin}}}(\bm{\nu},f):=\kappa_{[1..r]}^{(q)}(\bm{u}(\operatorname{LLT}^{\operatorname{cospin}})),\hskip
5.69046pt$ $\displaystyle\text{where }\hskip
5.69046ptu(\operatorname{LLT}^{\operatorname{cospin}})_{B}:=\operatorname{LLT}^{\operatorname{cospin}}(\bm{\nu},f)^{B},$
(11)
$\displaystyle\kappa_{\operatorname{LLT}^{\operatorname{Mac}}}(\bm{\nu},f):=\kappa_{[1..r]}^{(q)}(\bm{u}(\operatorname{LLT}^{\operatorname{Mac}})),\hskip
5.69046pt$ $\displaystyle\text{where }\hskip
5.69046ptu(\operatorname{LLT}^{\operatorname{Mac}})_{B}:=\operatorname{LLT}^{\operatorname{Mac}}(\bm{\nu},f)^{B},$
(12)
$\displaystyle\kappa_{\operatorname{LLT}}(\bm{\nu},f):=\kappa_{[1..r]}^{(q)}(\bm{u}(\operatorname{LLT})),\hskip
5.69046pt$ $\displaystyle\text{where }\hskip
5.69046ptu(\operatorname{LLT})_{B}:=\operatorname{LLT}(\bm{\nu},f)^{B}.$

Note that for any $\ell$-tuple of skew Young diagrams $\bm{\nu}$ there exists
a unique $1$-colored tuple of skew Young diagrams
$(\bm{\nu},\pi^{[1..\ell]}_{[1]})$, where $\pi^{[1..\ell]}_{[1]}$ is the
unique surjection of $[1..\ell]$ onto $\\{1\\}$. In this case, the cumulants
$\kappa_{\operatorname{LLT}^{\operatorname{cospin}}}(\bm{\nu},\operatorname{id}_{[1..\ell]}),\kappa_{\operatorname{LLT}^{\operatorname{Mac}}}(\bm{\nu},\operatorname{id}_{[1..\ell]})$,
and $\kappa_{\operatorname{LLT}}(\bm{\nu},\operatorname{id}_{[1..\ell]})$
coincide with the associated LLT functions
$\operatorname{LLT}^{\operatorname{cospin}}(\bm{\nu}),\operatorname{LLT}^{\operatorname{Mac}}(\bm{\nu})$,
and $\operatorname{LLT}(\bm{\nu})$, respectively. In general, LLT-cumulants
can be interpreted as an $r$-colored generalization of LLT polynomials.

The concept of $r$-colored tuples of skew shapes arose from the definition of
cumulants of the symmetric functions naturally indexed by partitions. This
definition was introduced in [DF17] (in the context of Jack and Macdonald
symmetric functions) as follows: let $\\{f_{\lambda}\\}$ be a class of
symmetric functions indexed by partitions. For partitions
$\lambda^{1},\dots,\lambda^{r}$, we define the family $(\bm{u})$ indexed by
subsets of $[1..r]$ as $u_{B}:=f_{\lambda^{B}}$, where the partition
$\lambda^{B}:=\bigoplus_{i\in B}\lambda^{i}$ is obtained from partitions
$\lambda^{i}\colon i\in B$ by summing their coordinates:
$\lambda^{B}_{j}:=\sum_{i\in B}\lambda^{i}_{j}$. We observe that the data of
partitions $\lambda^{1},\dots,\lambda^{r}$ can be alternatively encoded as an
$r$-colored partition $(\lambda=\lambda^{[1..r]},f)$ as follows: let $\lambda$
be a partition and let $f\colon[1..\ell(\lambda)]\to[1..r]$ be a surjective
function (that we interpret as the coloring of columns of the Young diagram
$\lambda$ by $r$ colors) such that the Young diagram formed by columns colored
by $i$ is equal to $\lambda^{i}$. Then, it is clear that for every
$B\subset[1..r]$, the Young diagram formed by columns colored by colors in $B$
is equal to $\lambda^{B}$. Of course, there are many colorings
$f\colon[1..\ell(\lambda)]\to[1..r]$ which encode partitions
$\lambda^{1},\dots,\lambda^{r}$ as an $r$-colored partition $(\lambda,f)$, but
among them there is a canonical choice, which we call the _canonical coloring_
$(\lambda,f_{\operatorname{cc}}\colon[1..\ell(\lambda)]\to[1..r])$. It is
uniquely determined by the following property: for any $i<j$ such that
$\lambda^{\prime}_{i}=\lambda^{\prime}_{j}$ (we recall that $\lambda^{\prime}$
denotes the _transpose_ of $\lambda$, i.e., the diagram with $\lambda_{1}$
boxes in the first column, $\lambda_{2}$ boxes in the second column, etc.),
one has $f_{\operatorname{cc}}(i)\leq f_{\operatorname{cc}}(j)$. This property
simply means that the Young diagram $\lambda^{[r]}$ can be obtained by sorting
the columns of $\lambda^{1},\dots,\lambda^{r}$ such that all the columns of
the same height are ordered with respect to the natural order $1<\cdots<r$,
see Fig. 1.

Figure 1. $r$-colored and canonically colored partitions.

When $f_{\lambda}=\tilde{H}_{\lambda}^{(q,t)}$ is the transformed version of
the Macdonald polynomial indexed by a partition $\lambda$, the corresponding
$q$-partial cumulant $\kappa_{[1..r]}^{(q)}(\bm{u})$ is called the _Macdonald
cumulant_ and denoted $\kappa(\lambda^{1},\dots,\lambda^{r})(\bm{x};q,t)$:

(13)
$\kappa(\lambda^{1},\dots,\lambda^{r})(\bm{x};q,t):=\kappa_{[1..r]}^{(q)}(\bm{u}).$

It was studied in [Doł17, Doł19], where its polynomiality and combinatorial
interpretation was obtained, generalizing the celebrated HHL formula (5).
Furthermore, it was conjectured in [Doł17] that Macdonald cumulants are Schur-
positive:

######  Conjecture 2.6 ([Doł17]).

Let $\lambda^{1},\dots,\lambda^{r}$ be partitions. Then for any partition
$\nu$ the coefficient $[s_{\nu}]\kappa(\lambda^{1},\dots,\lambda^{r})$ in the
Schur expansion of the Macdonald cumulant is a polynomial in $q,t$ with
nonnegative coefficients.

The first motivation for introducing LLT cumulants is to attack 2.6. Since
Macdonald polynomials can be naturally decomposed into LLT polynomials, it is
natural to ask whether a similar decomposition occurs for Macdonald cumulants.
Moreover, it was proved by Grojnowski and Haiman [GH07] that LLT polynomials
are Schur-positive, which gives an alternative proof of the Schur positivity
of Macdonald polynomials. Extensive computer simulations performed by the
authors using the SageMath computer algebra system [The20] tend us to believe
that the result of Grojnowski and Haiman might be a special case of Schur-
positivity that holds for the more general class of LLT cumulants. Therefore,
we propose the following conjecture:

######  Conjecture 2.7.

For any $r$-colored tuple of skew shapes $(\bm{\nu},f)$ and for any partition
$\lambda$ the coefficient
$[s_{\lambda}]\kappa_{\operatorname{LLT}^{\operatorname{cospin}}}(\bm{\nu},f)$
in the Schur expansion of the LLT cumulant is a polynomial in $q$ with
nonnegative integer coefficients.

######  Example.

Let $\bm{\nu}=((2,2)/(1),(2),(1,1))$ be a tuple of three skew shapes. Consider
two colorings $f\colon[1..3]\to[1..3]$ and $f^{\prime}\colon[1..3]\to[1..2]$
defined by $f(i)=i$ for $1\leq i\leq 3$ and
$f^{\prime}(1)=f^{\prime}(3)=1,f^{\prime}(2)=2$. The corresponding cumulants
$\kappa_{\operatorname{LLT}^{\operatorname{cospin}}}(\bm{\nu},f)$ and
$\kappa_{\operatorname{LLT}^{\operatorname{cospin}}}(\bm{\nu},f^{\prime})$ are
equal to

$\displaystyle\kappa_{\operatorname{LLT}^{\operatorname{cospin}}}(\bm{\nu},f)$
$\displaystyle=(q-1)^{-2}\bigg{(}\operatorname{LLT}^{\operatorname{cospin}}((2,2)/(1),(2),(1,1))-\operatorname{LLT}^{\operatorname{cospin}}((2,2)/(1))\cdot$
$\displaystyle\cdot\operatorname{LLT}^{\operatorname{cospin}}((2),(1,1))-\operatorname{LLT}^{\operatorname{cospin}}((2))\operatorname{LLT}^{\operatorname{cospin}}((2,2)/(1),(1,1))$
$\displaystyle-\operatorname{LLT}^{\operatorname{cospin}}((1,1))\operatorname{LLT}^{\operatorname{cospin}}((2,2)/(1),(2))$
$\displaystyle+2\operatorname{LLT}^{\operatorname{cospin}}((2,2)/(1))\operatorname{LLT}^{\operatorname{cospin}}((2))\operatorname{LLT}^{\operatorname{cospin}}((1,1))\bigg{)},$
$\displaystyle\kappa_{\operatorname{LLT}^{\operatorname{cospin}}}(\bm{\nu},f^{\prime})$
$\displaystyle=(q-1)^{-1}\bigg{(}\operatorname{LLT}^{\operatorname{cospin}}((2,2)/(1),(2),(1,1))$
$\displaystyle-\operatorname{LLT}^{\operatorname{cospin}}((2))\operatorname{LLT}^{\operatorname{cospin}}((2,2)/(1),(1,1))\bigg{)}.$

Expanding them in the basis of Schur functions we have

$\displaystyle\kappa_{\operatorname{LLT}^{\operatorname{cospin}}}(\bm{\nu},f)$
$\displaystyle=(q^{2}+2q+2)s_{(2,2,1,1,1)}+(q+2)s_{(2,2,2,1)}$
$\displaystyle+(q^{2}+2q+2)s_{(3,1,1,1,1)}+(2q+4)s_{(3,2,1,1)}$
$\displaystyle+2s_{(3,2,2)}+s_{(3,3,1)}+(q+2)s_{(4,1,1,1)}+s_{(4,2,1)},$
$\displaystyle\kappa_{\operatorname{LLT}^{\operatorname{cospin}}}(\bm{\nu},f^{\prime})$
$\displaystyle=(q^{3}+q^{2})s_{(2,2,1,1,1)}+(q^{2}+q)s_{(2,2,2,1)}$
$\displaystyle+(q^{3}+q^{2})s_{(3,1,1,1,1)}+(2q^{2}+2q)s_{(3,2,1,1)}+(2q+1)s_{(3,2,2)}$
$\displaystyle+(q+1)s_{(3,3,1)}+(q^{2}+q)s_{(4,1,1,1)}+(q+1)s_{(4,2,1)}.$

In the following, we prove that 2.7 implies 2.6. In order to do this, we
express Macdonald cumulants as a positive linear combination of LLT cumulants,
generalizing the classical decomposition from Theorem 2.2 to cumulants, and we
show that Schur-positivity of LLT cumulants can be put into the following
hierarchy: Schur-positivity of
$\kappa_{\operatorname{LLT}^{\operatorname{cospin}}}(\bm{\nu},f)$ implies
Schur-positivity of
$\kappa_{\operatorname{LLT}^{\operatorname{Mac}}}(\bm{\nu},f)$, which further
implies Schur-positivity of $\kappa_{\operatorname{LLT}}(\bm{\nu},f)$.

######  Remark 2.8.

In fact, the chain of implications mentioned above is valid only when we
restrict $\bm{\nu}$ to be a sequence of ribbon shapes due to the definition of
the normalization $\operatorname{LLT}^{\operatorname{Mac}}(\bm{\nu})$ (see
(4)). However, Schur-positivity of
$\kappa_{\operatorname{LLT}^{\operatorname{cospin}}}(\bm{\nu},f)$ for all
$r$-colored tuples of skew-shapes $(\bm{\nu},f)$ implies Schur-positivity of
$\kappa_{\operatorname{LLT}}(\bm{\nu},f)$ for all $r$-colored tuples of skew-
shapes $(\bm{\nu},f)$, which will be clear from the proof of Theorem 2.10.

####  2.3.1. Decomposition of Macdonald cumulants

######  Theorem 2.9.

Let $\lambda^{1},\dots,\lambda^{r}$ be partitions. Then, the following
identity holds true:

(14)
$\kappa(\lambda^{1},\dots,\lambda^{r})(\bm{x};q,t)=\sum_{\bm{\nu}}t^{\operatorname{maj}(\bm{\nu})}\kappa_{\operatorname{LLT}^{\operatorname{Mac}}}(\bm{\nu},f_{\operatorname{cc}}),$

where we sum over all tuples of ribbons whose bottom, far-right cell has
content $0$ and such that $|\bm{\nu}_{j}|=(\lambda^{[1..r]})^{\prime}_{j}$ for
$1\leq j\leq\ell(\lambda^{[1..r]})$ (i.e. the size of the $j$-th ribbon is
equal to the length of the $j$-th column of $\lambda^{[1..r]}$) and
$f_{\operatorname{cc}}$ is the canonical coloring associated with
$\lambda^{1},\dots,\lambda^{r}$.

###### Proof.

It is a direct consequence of the interpretation of the Macdonald cumulant as
the $q$-partial cumulant of the canonically $r$-colored partition and of
Theorem 2.2. Indeed, note that for any subset $B\subset[1..r]$, Theorem 2.2
applied to $\lambda=\lambda^{B}$ gives

$\tilde{H}_{\lambda^{B}}^{(q,t)}=\sum_{\bm{\nu}}t^{\operatorname{maj}(\bm{\nu})}\operatorname{LLT}^{\operatorname{Mac}}(\bm{\nu}),$

where we sum over skew-partitions whose $j$-th element is a ribbon of length
$(\lambda^{B})^{\prime}_{j}$ whose bottom, far-right cell has content $0$. In
particular, for any set-partition $\pi\in\mathcal{P}([1..r])$, one has

$\prod_{B\in\pi}\tilde{H}_{\lambda^{B}}^{(q,t)}=\sum_{\bm{\nu}}\prod_{B\in\pi}t^{\operatorname{maj}((\bm{\nu},f_{\operatorname{cc}})^{B})}\operatorname{LLT}^{\operatorname{Mac}}((\bm{\nu},f_{\operatorname{cc}})^{B}),$

where the sum runs over the same set as the summation in (5).

Strictly from the definition (6) of $\operatorname{maj}$, one has
$\sum_{B\in\pi}\operatorname{maj}((\bm{\nu},f_{\operatorname{cc}})^{B})=\operatorname{maj}(\bm{\nu})$
for any set-partition $\pi\in\mathcal{P}([1..r])$, and formula (14) follows. ∎

######  Theorem 2.10.

Suppose that 2.7 holds true. Then, for any $r$-colored tuple of skew shapes
$(\bm{\nu},f)$ and for any partition $\nu$, the coefficients

$[s_{\nu}]\kappa_{\operatorname{LLT}^{\operatorname{Mac}}}(\bm{\nu},f)\in\mathbb{Z}_{\geq
0}[q],\qquad[s_{\nu}]\kappa_{\operatorname{LLT}}(\bm{\nu},f)\in\mathbb{Z}_{\geq
0}[q]$

are polynomials in $q$ with nonnegative integer coefficients. In particular,
2.6 holds true.

###### Proof.

Recall the definiton Eq. 10 of LLT cumulants. We will show that there exist
graphs $G\subset G^{\prime}$ such that

(15) $\displaystyle\kappa_{[1..r]}^{(q)}(\bm{u}(\operatorname{LLT}))$
$\displaystyle=\kappa_{[1..r]}^{(q)}(\bm{u}(\operatorname{LLT}^{\operatorname{cospin}})^{G^{\prime}}),$
(16)
$\displaystyle\kappa_{[1..r]}^{(q)}(\bm{u}(\operatorname{LLT}^{\operatorname{Mac}}))$
$\displaystyle=\kappa_{[1..r]}^{(q)}(\bm{u}(\operatorname{LLT}^{\operatorname{cospin}})^{G}).$

Then the statements follow directly from Theorem 2.5 and Eq. 14.

Notice that the family of nonnegative integers $(e_{B})_{B\subset V}$ indexed
by subsets of the set $V$ is the number of edges in some graph $G=(V,E)$
linking vertices in $B$ if and only if

(17) $e_{B}\geq\sum_{b\in B}e_{\\{b\\}}\ \ \ \text{ and }\ \ \
e_{B}=\sum_{\begin{subarray}{c}B^{\prime}\subset B,\\\
|B^{\prime}|=2\end{subarray}}e_{B^{\prime}}-(|B|-2)\sum_{b\in B}e_{\\{b\\}}.$

Indeed, the inequality corresponds to $e_{B}$ counting all the loops on
vertices from $B$, and the equality counts the edges between each pair of
vertices from $B$ minus the overcounted loops.

We first prove that there exist graphs $G,G^{\prime}$ such that (15) and (16)
hold. To show (15), consider
$e_{B}=\min_{T\in\operatorname{SSYT}((\bm{\nu},f)^{B})}\operatorname{inv}(T)=|\operatorname{Inv}(T)\setminus\operatorname{Inv}_{\operatorname{cospin}}(T)|$,
which does not depend on the choice of
$T\in\operatorname{SSYT}((\bm{\nu},f)^{B})$. Then the conditions in (17) are
satisfied since for a pair of boxes $\square\in(\bm{\nu},f)^{\\{i\\}}$ and
$\square^{\prime}\in(\bm{\nu},f)^{\\{j\\}}$, one has
$(\square,\square^{\prime})\in\operatorname{Inv}(T)\setminus\operatorname{Inv}_{\operatorname{cospin}}(T)$
if and only if
$(\square,\square^{\prime})\in\operatorname{Inv}(T_{\\{i,j\\}})\setminus\operatorname{Inv}_{\operatorname{cospin}}(T_{\\{i,j\\}})$,
where $T_{\\{i,j\\}}$ is a tableau $T$ restricted to
$(\bm{\nu},f)^{\\{i,j\\}}$.

Similarly, for

$e_{B}=a((\bm{\nu},f)^{B})=\sum_{\square\in\operatorname{Des}((\bm{\nu},f)^{B})}|\\{\square^{\prime}:c(\square^{\prime})=c(\square),\tilde{c}(\square^{\prime})>\tilde{c}(\square)\\}|,$

one has

(18)
$\sum_{\square\in\operatorname{Des}((\bm{\nu},f)^{B})}|\\{\square^{\prime}\colon
c(\square^{\prime})=c(\square),\tilde{c}(\square^{\prime})>\tilde{c}(\square)\\}|=\\\
\sum_{b\in
B}\sum_{\square\in\operatorname{Des}((\bm{\nu},f)^{\\{b\\}})}\sum_{b^{\prime}\in
B}|\\{\square^{\prime}\in(\bm{\nu},f)^{\\{b^{\prime}\\}}\colon
c(\square^{\prime})=c(\square),\tilde{c}(\square^{\prime})>\tilde{c}(\square)\\}|.$

Note that for any subset $A\subset B$ and for each pair of boxes
$(\square,\square^{\prime})\in(\bm{\nu},f)^{A}$, there is a uniquely
associated pair of boxes $(\square,\square^{\prime})\in(\bm{\nu},f)^{B}$ and
their contents are identical in $(\bm{\nu},f)^{A}$ and $(\bm{\nu},f)^{B}$,
while their shifted contents might be different but the relation
$\tilde{c}(\square^{\prime})>\tilde{c}(\square)$ is again the same in both
$(\bm{\nu},f)^{A}$ and $(\bm{\nu},f)^{B}$. This observation together with (18)
implies that the quantities $e_{B}$ satisfy (17). This proves (16).

Finally, we prove that $G\subset G^{\prime}$, which is equivalent to proving
that
$a((\bm{\nu},f)^{B})\leq\min_{T\in\operatorname{SSYT}((\bm{\nu},f)^{B})}\operatorname{inv}(T)$.
Let $\square\in\operatorname{Des}((\bm{\nu},f)^{B})$ and
$\square^{\prime}\in(\bm{\nu},f)^{B}$ be such that
$c(\square^{\prime})=c(\square),\tilde{c}(\square^{\prime})>\tilde{c}(\square)$.
For any $T\in\operatorname{SSYT}((\bm{\nu},f)^{B})$, we necessarily have
$T(\square)>T(\square_{\downarrow})$. Therefore, either
$(\square,\square^{\prime})\in\operatorname{Inv}(T)$ or
$(\square^{\prime},\square_{\downarrow})\in\operatorname{Inv}(T)$ (or both).
Summing over all $\square\in\operatorname{Des}((\bm{\nu},f)^{B})$ and

$\square^{\prime}\in\\{\square^{\prime}\in(\bm{\nu},f)^{B}:c(\square^{\prime})=c(\square),\tilde{c}(\square^{\prime})>\tilde{c}(\square)\\},$

we have that $a((\bm{\nu},f)^{B})\leq\operatorname{inv}(T)$ for any
$T\in\operatorname{SSYT}((\bm{\nu},f)^{B})$. Thus,
$a((\bm{\nu},f)^{B})\leq\min_{T\in\operatorname{SSYT}((\bm{\nu},f)^{B})}\operatorname{inv}(T)$,
which is equivalent to the fact that $G\subset G^{\prime}$. This implies that
$[s_{\nu}]\kappa_{\operatorname{LLT}^{\operatorname{Mac}}}(\bm{\nu},f)$ and
$[s_{\nu}]\kappa_{\operatorname{LLT}}(\bm{\nu},f)$ are indeed polynomials in
$q$, which finishes the proof. ∎

##  3\. Graph colorings

In the following, we interpret LLT polynomials as the generating functions of
colorings of certain directed graphs. This viewpoint provides a natural graph-
theoretic interpretation of LLT cumulants as well as various positivity
properties generalizing some recent results [AS22, Doł19].

###  3.1. LLT graphs and cumulants of $r$-colored LLT graphs

######  Definition 3.1.

We call $G$ an _LLT graph_ if it is a finite directed graph with three types
of edges, visually depicted as $\rightarrow$, $\twoheadrightarrow$, and
$\Rightarrow$, which we call _edges of type I_ , _of type II_ , and _double
edges_ , respectively. Denote the corresponding sets of edges by $E_{1}(G)$,
$E_{2}(G)$, and $E_{d}(G)$. Additionally, write $\mathscr{G}$ for the
$\mathbb{Z}[q]$-module spanned by LLT graphs and $\mathscr{G}_{1}<\mathscr{G}$
for the submodule generated by LLT graphs with only edges of type II
($E_{1}(G)=E_{d}(G)=\emptyset$).

Let $\operatorname{QSym}$ denote the ring of quasi-symmetric functions over
$\mathbb{Z}[q]$. We recall that a quasisymmetric function $f$ is a power
series in variables $x_{1},x_{2},\dots$ of a bounded degree such that for any
sequence of positive integers $(\alpha_{1},\dots,\alpha_{n})$ the coefficients
of the monomial $[x_{i_{1}}^{\alpha_{1}}\cdots x_{i_{n}}^{\alpha_{n}}]f$ in
$f$ is the same for all possible choices of indices $i_{1}<\dots<i_{n}$ (see
[Ges84] for more details on $\operatorname{QSym}$). With an LLT graph $G$ we
associate its _LLT polynomial_ :

(19)
$\operatorname{LLT}(G):=\sum\limits_{f:V(G)\rightarrow\mathbb{N}}\left(\prod\limits_{(u,v)\in
E(G)}\varphi_{f}(u,v)\right)\cdot\left(\prod\limits_{v\in
V(G)}x_{f(v)}\right),$

with

(20) $\varphi_{f}(u,v)=\begin{cases}[f(u)>f(v)]&\text{for }(u,v)\in
E_{1}(G);\\\ [f(u)\geq f(v)]&\text{for }(u,v)\in E_{2}(G);\\\
q[f(u)>f(v)]+[f(u)\leq f(v)]&\text{for }(u,v)\in E_{d}(G),\end{cases}$

where $[A]$ is the characteristic function of condition $A$, i.e., is equal to
$1$ if $A$ is true and $0$ otherwise.

There is an obvious way to associate an LLT graph $G_{\bm{\nu}}$ to a sequence
of skew shapes $\bm{\nu}$ such that
$\operatorname{LLT}(G_{\bm{\nu}})=\operatorname{LLT}(\bm{\nu})$. To be
precise, vertices correspond to cells, edges of type I go from a cell
$\square$ to $\square_{\downarrow}$, edges of type II go from a cell $\square$
to $\square_{\leftarrow}$, and double edges connect cells that correspond to
inversions (see Fig. 2).

$a$$b$$c$$d$$x$$y$

$a$$b$$c$$d$$x$$y$

Figure 2. The LLT graph corresponding to ((3,2)/(1), (1,1)).

Let $G$ be an LLT graph and let $\vec{e_{i}}\in E_{i}(G)$ for $i\in\\{1,d\\}$.
Define the local transformation

$\pi_{\vec{e_{i}}}(G)=\begin{cases}G\setminus\\{\vec{e_{1}}\\}-G_{\vec{e_{1}}\to\cev{e_{2}}}&\text{
for }i=1,\\\
qG\setminus\\{\vec{e_{d}}\\}+(1-q)G_{\vec{e_{d}}\to\cev{e_{2}}}&\text{ for
}i=d,\end{cases}$

where $\vec{e_{i}}\to\cev{e_{j}}$ ($\vec{e_{i}}\to\vec{e_{j}}$, respectively)
denotes replacing the directed edge $\vec{e_{i}}$ of type $i$ by the edge of
type $j$ with the opposite (the same, respectively) direction.

######  Example.

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We define

$\pi(G):=\bigg{(}\prod_{\vec{e}\in E_{1}(G)\cup
E_{d}(G)}\pi_{\vec{e}}\bigg{)}(G)$

as the concatenation of local transformations over all edges of type I and
double edges (these transformations are commutative so their order does not
matter and this concatenation is well-defined). Note that local
transformations kill all edges of type I and $d$ and thus, the map
$\pi\colon\mathscr{G}\to\mathscr{G}_{1}$ is well-defined. In fact, we claim
that the map $\operatorname{LLT}\colon\mathscr{G}\to\operatorname{QSym}$ is a
well-defined surjective homomorphism such that
$\operatorname{LLT}(G)=\operatorname{LLT}\circ\pi(G)$ for every LLT graph.

######  Lemma 3.2.

For $\mathscr{G}$ and $\mathscr{G}_{1}$ as in Definition 3.1, the following
diagram is commutative:

${\mathscr{G}}$${\mathscr{G}_{1}}$${\operatorname{QSym}}$$\scriptstyle{\pi}$$\scriptstyle{\operatorname{LLT}}$$\scriptstyle{\operatorname{LLT}}$

###### Proof.

Let $G$ be an LLT graph. It was proved by Féray [F1́5] that
$\operatorname{LLT}\colon\mathscr{G}_{1}\to\operatorname{QSym}$ is a well-
defined surjective homomorphism. Moreover, it is straightforward from the
definition of the map $\operatorname{LLT}$ that it is invariant under the
local transformations, i.e., for every $\vec{e}\in E_{1}(G)\cup E_{d}(G)$, one
has $\operatorname{LLT}(\pi_{\vec{e}}(G))=\operatorname{LLT}(G)$. Thus,
$\operatorname{LLT}(G)=\operatorname{LLT}(\pi(G))$, which finishes the proof.
∎

######  Remark 3.3.

Let $\widehat{\mathscr{G}}_{1}<\mathscr{G}_{1}$ be a submodule of
$\mathscr{G}_{1}$ spanned by acyclic graphs. The main result of Féray [F1́5]
is an explicit description of the kernel of the map
$\operatorname{LLT}\colon\widehat{\mathscr{G}}_{1}\to\operatorname{QSym}$ by
using the _cyclic inclusion-exclusion principle_. This description together
with Lemma 3.2 can be a priori used to describe the kernel of the morphism
$\operatorname{LLT}\colon\mathscr{G}\to\operatorname{QSym}$, thus to
understand all the relations between LLT graphs under the $\operatorname{LLT}$
morphism. Additionally, $\mathscr{G}$ seems to carry a natural Hopf algebra
structure. Studying various relations between LLT polynomials is a very active
topic recently and it proved to be useful in understanding the combinatorial
structure of LLT polynomials [Lee21, HNY20, AN21, AS22, Tom21]. We believe
that further studies in the direction of understanding the algebraic structure
of the pair $(\mathscr{G},\operatorname{LLT})$ might bring better
understanding of the combinatorial structure of LLT polynomials, and we leave
this problem for future research.

As a consequence of Lemma 3.2 and its proof, we obtain two identities
expressing the LLT polynomial of a given LLT graph $G$ in terms of two
important LLT graphs, which do not have any double edges. For any subset
$E\subset E_{d}(G)$, we define $G^{E}$ and $\tilde{G}^{E}$ as follows:

  * •

$V(\tilde{G}^{E})=V(G^{E})=V(G)$,

  * •

$E_{d}(\tilde{G}^{E})=E_{d}(G^{E})=\emptyset$,

  * •

$E_{1}(\tilde{G}^{E})=E_{1}(G^{E})=E_{1}(G)\cup E$,

  * •

$E_{2}(\tilde{G}^{E})=E_{2}(G)\cup\\{(u,v)\mid(v,u)\in E_{d}\setminus E\\}$,
and $E_{2}(G^{E})=E_{2}(G)$.

######  Example.

For

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}\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{45.52458pt}{19.74226pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{2.20001pt}\pgfsys@invoke{
}{}\pgfsys@moveto{45.52458pt}{0.0pt}\pgfsys@lineto{0.0pt}{45.52458pt}\pgfsys@stroke\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{
}{\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@moveto{45.52458pt}{0.0pt}\pgfsys@lineto{0.0pt}{45.52458pt}\pgfsys@stroke\pgfsys@invoke{
}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{2.20001pt}\pgfsys@invoke{
}{}{}{}{}{{}}\pgfsys@moveto{45.52458pt}{0.0pt}\pgfsys@lineto{24.89775pt}{20.62682pt}\pgfsys@stroke\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{
}{\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@moveto{45.52458pt}{0.0pt}\pgfsys@lineto{24.89775pt}{20.62682pt}\pgfsys@stroke\pgfsys@invoke{
}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.7071}{0.7071}{-0.7071}{-0.7071}{24.89777pt}{20.6268pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{24.71472pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{20.55481pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{24.71472pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{27.31467pt}{0.0pt}\pgfsys@lineto{45.52458pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}{}{}{}{}{{}}\pgfsys@moveto{45.52458pt}{45.52458pt}\pgfsys@lineto{20.80983pt}{45.52458pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{24.96974pt}{45.52458pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{20.80983pt}{45.52458pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{18.20988pt}{45.52458pt}\pgfsys@lineto{0.0pt}{45.52458pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}{}{}{}{{}}\pgfsys@moveto{45.52458pt}{0.0pt}\pgfsys@lineto{65.68694pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{65.68694pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{68.28688pt}{0.0pt}\pgfsys@lineto{91.04916pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}\pgfsys@color@rgb@stroke{0.75}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0.75}{0}{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.75,0,0}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{2.20001pt}\pgfsys@invoke{
}{}\pgfsys@moveto{91.04916pt}{0.0pt}\pgfsys@lineto{45.52458pt}{45.52458pt}\pgfsys@stroke\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{
}{\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@moveto{91.04916pt}{0.0pt}\pgfsys@lineto{45.52458pt}{45.52458pt}\pgfsys@stroke\pgfsys@invoke{
}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}\pgfsys@color@rgb@stroke{0.75}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0.75}{0}{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.75,0,0}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{2.20001pt}\pgfsys@invoke{
}{}{}{}{}{{}}\pgfsys@moveto{91.04916pt}{0.0pt}\pgfsys@lineto{70.42235pt}{20.62682pt}\pgfsys@stroke\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{
}{\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@moveto{91.04916pt}{0.0pt}\pgfsys@lineto{70.42235pt}{20.62682pt}\pgfsys@stroke\pgfsys@invoke{
}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.7071}{0.7071}{-0.7071}{-0.7071}{70.42236pt}{20.6268pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}$

and $E\subset E_{d}(G)$ equal to the set of the red edges above, we have

$G^{E}=\vbox{\hbox{ \leavevmode\hbox to91.85pt{\vbox
to46.32pt{\pgfpicture\makeatletter\hbox{\hskip 0.4pt\lower-0.4pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{20.16234pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{0.0pt}{20.16234pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{22.76228pt}\pgfsys@lineto{0.0pt}{45.52458pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{45.52458pt}{0.0pt}\pgfsys@lineto{45.52458pt}{45.52458pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}{}{}{}{{}}\pgfsys@moveto{45.52458pt}{0.0pt}\pgfsys@lineto{45.52458pt}{20.16234pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{45.52458pt}{20.16234pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{24.71472pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{20.55481pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{24.71472pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{27.31467pt}{0.0pt}\pgfsys@lineto{45.52458pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}{}{}{}{}{{}}\pgfsys@moveto{45.52458pt}{45.52458pt}\pgfsys@lineto{20.80983pt}{45.52458pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{24.96974pt}{45.52458pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{20.80983pt}{45.52458pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{18.20988pt}{45.52458pt}\pgfsys@lineto{0.0pt}{45.52458pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}{}{}{}{{}}\pgfsys@moveto{45.52458pt}{0.0pt}\pgfsys@lineto{65.68694pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{65.68694pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{68.28688pt}{0.0pt}\pgfsys@lineto{91.04916pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{91.04916pt}{0.0pt}\pgfsys@lineto{45.52458pt}{45.52458pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}{}{}{}{{}}\pgfsys@moveto{91.04916pt}{0.0pt}\pgfsys@lineto{70.12532pt}{20.92384pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.7071}{0.7071}{-0.7071}{-0.7071}{70.12532pt}{20.92384pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}},\hskip
71.13188pt\tilde{G}^{E}=\vbox{\hbox{ \leavevmode\hbox to91.85pt{\vbox
to46.32pt{\pgfpicture\makeatletter\hbox{\hskip 0.4pt\lower-0.4pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{20.16234pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{0.0pt}{20.16234pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{22.76228pt}\pgfsys@lineto{0.0pt}{45.52458pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{45.52458pt}{0.0pt}\pgfsys@lineto{45.52458pt}{45.52458pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}{}{}{}{{}}\pgfsys@moveto{45.52458pt}{0.0pt}\pgfsys@lineto{45.52458pt}{20.16234pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{45.52458pt}{20.16234pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{45.52458pt}{0.0pt}\pgfsys@lineto{0.0pt}{45.52458pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{45.52458pt}\pgfsys@lineto{25.47623pt}{20.04832pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.7071}{-0.7071}{0.7071}{0.7071}{22.53473pt}{22.98982pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.7071}{-0.7071}{0.7071}{0.7071}{25.47623pt}{20.04832pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{24.71472pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{20.55481pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{24.71472pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{27.31467pt}{0.0pt}\pgfsys@lineto{45.52458pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}{}{}{}{}{{}}\pgfsys@moveto{45.52458pt}{45.52458pt}\pgfsys@lineto{20.80983pt}{45.52458pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{24.96974pt}{45.52458pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
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######  Corollary 3.4.

For any LLT graph $G$, we have

$\operatorname{LLT}(G)=\sum_{E\subseteq
E_{d}(G)}q^{|E|}\operatorname{LLT}(\tilde{G}^{E})=\sum_{E\subseteq
E_{d}(G)}(q-1)^{|E|}\operatorname{LLT}(G^{E}).$

###### Proof.

Note that

$\bigg{(}\prod_{\vec{e}\in
E_{d}(G)}\pi^{\prime}_{\vec{e}}\bigg{)}(G)=\sum_{E\subseteq
E_{d}(G)}q^{|E|}\tilde{G}^{E},\ \ \bigg{(}\prod_{\vec{e}\in
E_{d}(G)}\pi^{\prime\prime}_{\vec{e}}\bigg{)}(G)=\sum_{E\subseteq
E_{d}(G)}(q-1)^{|E|}G^{E},$

where
$\pi^{\prime}_{\vec{e_{d}}}(G)=G_{\vec{e_{d}}\to\cev{e_{2}}}+qG_{\vec{e_{d}}\to\vec{e_{1}}}$
and
$\pi^{\prime\prime}_{\vec{e_{d}}}(G)=G\setminus\\{\vec{e_{d}}\\}+(q-1)G_{\vec{e_{d}}\to\vec{e_{1}}}$
for $e_{d}\in E_{d}(G)$. Moreover,
$\operatorname{LLT}(G)=\operatorname{LLT}(\pi^{\prime}_{\vec{e_{d}}}(G))$
follows from the definition, and
$\operatorname{LLT}(G)=\operatorname{LLT}(\pi^{\prime\prime}_{\vec{e_{d}}}(G))$
follows from

$\operatorname{LLT}(G)=\operatorname{LLT}(G_{\vec{e_{d}}\to\cev{e_{2}}})+q\operatorname{LLT}(G_{\vec{e_{d}}\to\vec{e_{1}}})=\operatorname{LLT}(G\setminus\\{\vec{e_{d}}\\})+(q-1)\operatorname{LLT}(G_{\vec{e_{d}}\to\vec{e_{1}}})$

since we have that
$\operatorname{LLT}(G_{\vec{e_{d}}\to\cev{e_{2}}})=\operatorname{LLT}(G\setminus\\{\vec{e_{d}}\\})-\operatorname{LLT}(G_{\vec{e_{d}}\to\vec{e_{1}}})$.
∎

The definition of LLT cumulants of $r$-colored tuples of skew-shapes
generalizes naturally to the definition of LLT cumulants of $r$-colored LLT
graphs.

######  Definition 3.5.

We say that $(G,f)$ is an _$r$ -colored LLT graph_ if $G$ is an LLT graph and
$f\in V(G)\to[1..r]$ is a surjective coloring of vertices of $G$ such that
both endpoints of edges in $E_{1}(G)\cup E_{2}(G)$ have the same color. For
any subset $B\subset[1..r]$, we define the vertex set $V_{B}:=\\{v\in
V(G)\colon f(v)\in B\\}$ and for any subset $V^{\prime}\subset V(G)$, we
define $G|_{V}$ as the subgraph of $G$ obtained by restricting its set of
vertices to $V^{\prime}$. Then, we define the _LLT cumulant_ of an $r$-colored
LLT graph $(G,f)$ as the $q$-partial cumulant $\kappa^{(q)}_{[1..r]}(\bm{u})$
for the family defined by

$u_{B}:=\operatorname{LLT}(G|_{V_{B}}).$

Observe that the first equation in Corollary 3.4 is, in fact, a special case
of a more general formula:

######  Corollary 3.6.

For any set-partition $\pi\in\mathcal{P}([1..r])$, one has

(21) $\prod_{B\in\pi}\operatorname{LLT}(G|_{V_{B}})=\sum_{E\subseteq
E_{d}(G)}\operatorname{LLT}(\tilde{G}^{E})\prod_{B\in\pi}q^{|E_{B}|},$

where $E_{B}\subset E$ is the subset of edges with both endpoints in $B$.

###### Proof.

Formula (21) is proved similarly to Corollary 3.4, so we only sketch the
proof. Let $G_{\pi}:=\bigoplus_{B\in\pi}G|_{V_{B}}$, where $G_{1}\oplus G_{2}$
is a disjoint union of the LLT graphs $G_{1}$ and $G_{2}$. Then
$\prod_{B\in\pi}\operatorname{LLT}(G|_{V_{B}})=\operatorname{LLT}(G_{\pi})$.
Note that $G_{\pi}$ is obtained from $G$ by removing all the double edges
connecting vertices with colors lying in different blocks of $\pi$. Consider
two local transformations: $\pi^{\prime}_{\vec{e}}(G_{\pi})$ for $\vec{e}\in
E_{d}(G_{\pi})$ and
$\pi^{\prime\prime\prime}_{\vec{e}}(G_{\pi}):=(G_{\pi})_{\vec{e}\to\vec{e_{1}}}+(G_{\pi})_{\vec{e}\to\cev{e_{2}}}$
for any orientation $\vec{e}$ of $e\notin E(G_{\pi})$. Notice that

$\bigg{(}\prod_{\vec{e}\in E_{d}(G_{\pi})\atop\vec{\tilde{e}}\in
E_{d}(G)\setminus
E_{d}(G_{\pi})}\pi^{\prime}_{\vec{e}}\pi^{\prime\prime\prime}_{\vec{\tilde{e}}}\bigg{)}(G_{\pi})=\sum_{E\subseteq
E_{d}(G)}\tilde{G}^{E}\prod_{B\in\pi}q^{|E_{B}|}.$

Finally, recall that $\operatorname{LLT}$ is invariant under taking the local
transformation $\pi^{\prime}_{\vec{e}}$ and notice that
$\operatorname{LLT}(G)=\operatorname{LLT}(\pi^{\prime\prime\prime}_{\vec{e}}(G))$
for any orientation $\vec{e}$ of $e\notin E(G)$. This finishes the proof. ∎

In the following, we prove that the LLT cumulant of the $r$-colored LLT graph
$(G,f)$ can be naturally expressed as a sum of LLT polynomials of so-called
$f$-connected graphs.

######  Definition 3.7.

Let $(G,f)$ be an $r$-colored LLT graph. We say that it is _$f$ -connected_ if
the graph $G_{f}$ obtained from $G$ by identifying vertices of the same color
is connected. In other words, the graph $G$ is $f$-connected if for every pair
$i,j\in[1..r]$, there exists $i=i_{0}\neq i_{1}\cdots\neq i_{k}=j\in[1..r]$
and vertices $v_{0},\dots,v_{k}\in V(G)$ colored by $i_{0},\dots,i_{k}$
respectively such that $v_{i-1}$ is connected to $v_{i}$ for every $1\leq
i\leq k$.

Note that when $f$ is a bijection then the graph $G$ is $f$-connected if and
only if $G$ is connected, and if $f$ is a $1$-coloring then the condition of
being $f$-connected is empty (it is always satisfied). We have the following
combinatorial interpretation of an LLT cumulant of an $r$-colored LLT graph
$(G,f)$.

######  Theorem 3.8.

Let $(G,f)$ be an $r$-colored LLT graph and denote $E_{d}=E_{d}(G)$. Then:

(22)
$\displaystyle\kappa_{\operatorname{LLT}}(G,f)(q+1)=\sum\limits_{\begin{subarray}{c}E\subseteq
E_{d}\\\ G^{E}\text{
$f$-connected}\end{subarray}}q^{|E|-r+1}\operatorname{LLT}(G^{E})(q+1).$

Theorem 3.8 essentially shows the structure behind the, a priori, algebraic
definition of a cumulant: it kills all $f$-disconnected summands in the
expansion and preserves the $f$-connected ones. Furthermore, we note that we
formulate the statement with the polynomials evaluated at $q+1$ to highlight
the $\operatorname{LLT}$-positivity of the cumulant after the shift
$q\longmapsto q+1$: an operation that is also relevant in the context of the
$e$-positivity phenomenon (see Section 3.2.4).

###### Proof of Theorem 3.8.

We have

$\displaystyle\kappa_{\operatorname{LLT}}(G,f)(q+1):$
$\displaystyle=q^{1-r}\sum_{\pi\in\mathcal{P}([1..r])}(-1)^{|\pi|-1}(|\pi|-1)!\prod_{B\in\pi}\operatorname{LLT}(G|_{V_{B}})$
$\displaystyle=q^{1-r}\sum_{\pi\in\mathcal{P}([1..r])}(-1)^{|\pi|-1}(|\pi|-1)!\operatorname{LLT}\left(G_{\pi}\right),$

where we recall that $G_{\pi}:=\bigoplus_{B\in\pi}G|_{V_{B}}$.

By Corollary 3.4, for each $B\in\mathcal{P}([1..r])$, we get

(23) $\operatorname{LLT}\left(G_{\pi}\right)(q+1)=\sum_{E\subseteq
E_{d}(G_{\pi})}q^{|E|}\operatorname{LLT}(G_{\pi}^{E})(q+1)$

so that

(24)
$\displaystyle\kappa_{\operatorname{LLT}}(G,f)(q+1)=q^{1-r}\sum_{\pi\in\mathcal{P}([1..r])}(-1)^{|\pi|-1}(|\pi|-1)!\sum_{E\subseteq
E_{d}(G_{\pi})}q^{|E|}\operatorname{LLT}(G_{\pi}^{E})(q+1).$

Now consider an LLT graph $G^{\prime}=G_{\sigma}^{E}$ for some
$\sigma\in\mathcal{P}([1..r])$ and $E\subseteq E_{d}(G_{\sigma})$. For a fixed
$G^{\prime}$ of this form, pick $\sigma$ to be minimal, i.e. pick $\sigma$
such that for every block $B\in\sigma$, the graph $G^{\prime}|_{V_{B}}$ is
$f$-connected. Note that $G^{\prime}$ is $f$-connected if and only if
$\sigma=\\{[1..r]\\}$. We compute the contribution of the graph $G^{\prime}$
to the RHS of the formula (24).

Note that $G^{\prime}$ appears in a summand corresponding to a partition $\pi$
if and only if for every $B\in\sigma$, there exists $C\in\pi$ such that
$B\subseteq C$. This is known as the containment relation $\sigma\leq\pi$ on
the set of set-partitions. Therefore, we have

$\displaystyle[\operatorname{LLT}(G^{\prime})(q+1)]\kappa_{\operatorname{LLT}}(G,f)$
$\displaystyle=q^{|E|-r+1}\sum_{\sigma\leq\pi}(-1)^{|\pi|-1}(|\pi|-1)!=q^{|E|-r+1}\delta_{\sigma,\\{[1..r]\\}}.$

The last equality comes from the well-known fact that
$(-1)^{|\pi|-1}(|\pi|-1)!$ is equal to the Möbius function
$\mu(\pi,\\{[1..r]\\})$ on the poset of set-partitions
$(\mathcal{P}([1..r]),\leq)$ and the sum of the Möbius function
$\mu(\pi,\\{[1..r]\\})$ over the interval $\pi\in[\sigma,\\{[1..r]\\}]$ is
non-zero (and equal to $1$) only if $\sigma=\\{[1..r]\\}$ (see, e.g.,
[Wei35]). This finishes the proof as $\sigma=\\{[1..r]\\}$ if and only if
$G^{\prime}$ is $f$-connected. ∎

###  3.2. Various positivity results

The purpose of this section is to derive various combinatorial formulae for an
LLT cumulant of an $r$-colored LLT graph and proving certain positivity
results. We start by a quick review on $G$-inversion polynomials and their
different interpretations.

####  3.2.1. $G$-inversion polynomials and Tutte polynomials

Let $G$ be a multigraph (with possible multiedges and multiloops, as
previously) on the set of vertices $[1..r]$. We say that $T$ is a _spanning
tree_ of $G$ if it is a subgraph of $G$ with the same set of vertices $[1..r]$
and it is a tree (it is connected and has no cycles). A pair $(i,j)$ is called
an _inversion_ of a spanning tree $T$ of $G$ if $i,j\neq 1$ and if $i$ is an
ancestor of $j$ and $i>j$. An inversion $(i,j)$ is a _$\kappa$ -inversion_ if,
additionally, $j$ is adjacent to the parent of $i$ in $G$. A $G$-inversion
polynomial is a generating function of spanning trees of $G$ counted with
respect to the number of $\kappa$-inversions.

Let $\tilde{G}$ be a graph obtained from $G$ by replacing all multiple edges
by single ones. We recall that for any subset $B\subset V$ we denote the
number of edges linking vertices in $B$ by $e_{B}$. The _$G$ -inversion
polynomial_ is given by

(25) $\mathcal{I}_{G}(q)=q^{\text{number of loops in
}G}\sum_{T\subset\tilde{G}}q^{\kappa(T)}\prod_{\\{i,j\\}\in
T}[e_{\\{i,j\\}}(G)]_{q},$

where the sum runs over all spanning trees of $\tilde{G}$,

(26) $\kappa(T)=\sum_{\\{i,j\\}-\kappa-\text{inversion in
}T}e_{\\{\text{parent}(i),j\\}}(G),$

and we use the standard notation
$[n]_{q}:=\frac{q^{n}-1}{q-1}=1+q+\cdots+q^{n-1}$. As we already mentioned in
the introduction, $\mathcal{I}_{G}(q)=\operatorname{Tutte}(1,q)$, where
$\operatorname{Tutte}(x,y)$ is the Tutte polynomial of $G$ (a classical graph
invariant introduced by Tutte in [Tut54]):

(27) $\operatorname{Tutte}_{G}(x,y)=\sum_{H\subset G}(x-1)^{c(H)-1}\
(y-1)^{|E(H)|-|V|+c(H)}.$

The summation index above runs over all (possibly disconnected) subgraphs of
$G$, $c(H)$ denotes the number of connected components of $H$, and $E(H)$ is
the set of edges of $H$. In fact, we have the following lemma, which is
essentially due to Gessel [Ges95] and Josuat-Vergès [JV13] (see also [Doł19]
for treating both frameworks in the setting of multigraphs).

######  Lemma 3.9.

Let $G$ be a multigraph with the vertex set $V=[1..r]$ and let $\bm{u}$ be a
family indexed by subsets of $[1..r]$ defined as $u_{B}:=q^{e_{B}}$ for every
$B\subset[1..r]$. Then we have the following equalities between the generating
series:

(28) $\mathcal{I}_{G}(q)=\operatorname{Tutte}_{G}(1,q)=\kappa^{(q)}(\bm{u}).$

####  3.2.2. Monomial positivity

Here, we prove the following theorem implying positivity of LLT cumulants for
arbitrary $r$-colored LLT graphs in the quasi-symmetric monomial basis (this
is a refinement of the main result from [Doł19]):

######  Theorem 3.10.

Let $(G,f)$ be an $r$-colored LLT graph and denote $E_{d}=E_{d}(G)$. Then:

(29)
$\displaystyle\kappa_{\operatorname{LLT}}(G,f)(q)=\sum\limits_{\begin{subarray}{c}E\subseteq
E_{d}\\\ \hat{G}^{E}\text{
$f$-connected}\end{subarray}}\mathcal{I}_{(\hat{G}^{E})_{f}}(q)\operatorname{LLT}(\tilde{G}^{E})(q),$

where $\hat{G}^{E}$ is obtained from $\tilde{G}^{E}$ by removing all the edges
of type II (i.e. $E(\hat{G}^{E})=E(\tilde{G}^{E})\setminus
E_{2}(\tilde{G}^{E})$).

###### Proof.

Let $\hat{G}^{E}$ be a graph obtained from $\tilde{G}^{E}$ by removing all the
edges of type II and we recall that $(\hat{G}^{E})_{f}$ is a graph obtained
from $\hat{G}^{E}$ by identifying vertices of the same color, i.e. $v\sim w$
if $f(v)=f(w)$. Note that the vertex set of $(\hat{G}^{E})_{f}$ is equal to
$[1..r]$ and $|E_{B}|$ in the previous formula is equal to the number of edges
of $(\hat{G}^{E})_{f}$ with both endpoints belonging to $B$ (that we denote by
$e_{B}$ to be consistent with the previous notation). Therefore, following
(21), we end up with the formula

$\displaystyle\kappa_{\operatorname{LLT}}(G,f)$
$\displaystyle=(q-1)^{1-r}\sum_{E\subseteq
E_{d}(G)}\operatorname{LLT}(\tilde{G}^{E})\bigg{(}\sum_{\pi\in\mathcal{P}([1..r])}(-1)^{|\pi|-1}(|\pi|-1)!\prod_{B\in\pi}q^{|E_{B}|}\bigg{)}$
$\displaystyle=\sum_{E\subseteq
E_{d}(G)}\kappa^{(q)}_{[1..r]}(\bm{u})\operatorname{LLT}(\tilde{G}^{E}),$

where $u_{B}:=q^{e_{B}}$. This can be rewritten as

$\sum_{E\subseteq
E_{d}(G)}\mathcal{I}_{(\hat{G}^{E})_{f}}(q)\operatorname{LLT}(\tilde{G}^{E})(q)$

thanks to Lemma 3.9. Finally, $\mathcal{I}_{(\hat{G}^{E})_{f}}(q)=0$ whenever
$(\hat{G}^{E})_{f}$ is not connected (because disconnected graphs have no
spanning trees), which is the very definition of being $f$-connected for
$\hat{G}^{E}$. This finishes the proof. ∎

####  3.2.3. Fundamental quasisymmetric functions and 2.7 for hooks

For any non-negative integer $n$ and a subset $A\subset[n-1]$, we define the
_fundamental quasisymmetric function_ $F_{n,A}(\bm{x})$ to be the expression

$F_{n,A}(\bm{x}):=\sum_{\begin{subarray}{c}i_{1}\leq\dots\leq i_{n}\\\ j\in
A\Longrightarrow i_{j}<i_{j+1}\end{subarray}}x_{i_{1}}\dots x_{i_{n}}.$

We say that a tableau $T\in\operatorname{SSYT}(\bm{\nu})$ of a sequence
$\bm{\nu}$ with $|\bm{\nu}|=n$ is _standard_ if $T:\bm{\nu}\rightarrow[n]$ is
a bijection, and denote that fact by $T\in\operatorname{SYT}(\bm{\nu})$. We
also define the set of descents $\operatorname{Des}(T)$ of $T$ (note that this
is not the same as the set of descents of a tuple of skew shapes $\bm{\nu}$,
which appeared in the definition of Macdonald polynomials) as the set of
$i\in[1..n]$ such that $\tilde{c}(T^{-1}(i+1))<\tilde{c}(T^{-1}(i))$.

In [HHL05a], Haglund, Haiman and Loehr implicitly111instead of LLT polynomials
they expanded Macdonald polynomials into fundamental quasisymmetric functions,
but their arguments can be directly applied to LLT polynomials yielding (30)
proved the following formula for the expansion of LLT polynomials in the
fundamental quasisymmetric functions.

######  Theorem 3.11 ([HHL05a]).

For a sequence of skew shapes $\bm{\nu}$ with $|\bm{\nu}|=n$, we have

(30)
$\operatorname{LLT}(\bm{\nu})=\sum\limits_{T\in\operatorname{SYT}(\bm{\nu})}q^{\operatorname{inv}(T)}F_{n,\operatorname{Des}(T)}(\bm{x}).$

What is more, we can obtain a similar result in our language and notation.

######  Corollary 3.12.

For any $r$-colored tuple $(\bm{\nu},f)$ of size $n$ and for any set partition
$\pi\in\mathcal{P}([1..r])$, we have

(31)
$\prod_{B\in\pi}\operatorname{LLT}((\bm{\nu},f)^{B})=\sum\limits_{T\in\operatorname{SYT}(\bm{\nu})}q^{\operatorname{inv}_{\pi}(T)}F_{n,\operatorname{Des}(T)}(\bm{x}),$

where $\operatorname{inv}_{\pi}(T)$ denotes the number of inversions in $T$
with both boxes in the same block of $\pi$.

###### Proof.

The result is a straightforward application of the arguments used in [HHL05a].
∎

Applying the same proof as in Theorem 3.10 to (31), we obtain the following
result (see also [Doł19, Section 5] for an analogous argument applied to
Macdonald cumulants):

######  Theorem 3.13.

Let $(\bm{\nu},f)$ be an $r$-colored sequence of skew shapes of size $n$.
Then:

(32)
$\kappa_{\operatorname{LLT}}(\bm{\nu},f)(q)=\sum\limits_{T\in\operatorname{SYT}(\bm{\nu})}\sum\mathcal{I}_{(\widehat{G_{\bm{\nu}}}^{E^{T}})_{f}}(q)F_{n,\operatorname{Des}(T)}(\bm{x}),$

where the second sum runs over all subsets $E^{T}\subseteq
E_{d}(G_{\bm{\nu}})$ for which $\widehat{G_{\bm{\nu}}}^{E^{T}}$ is
$f$-connected and $T(i)>T(j)$ whenever $(i,j)\in
E(\widehat{G_{\bm{\nu}}}^{E^{T}})$.

In [Doł19], we were able to find an explicit formula for the coefficients of
Schur symmetric functions indexed by _hooks_ , i.e., partitions of the form
$(k,1^{n-k})$, in Macdonald cumulants, thanks to the arguments from [HHL05a].
Here, we will use a very nice theorem of Egge, Loehr and Warrington [ELW10]
which gives a combinatorial description of Schur coefficients of any symmetric
function when given an expansion in fundamental quasisymmetric functions.

######  Theorem 3.14 ([ELW10]).

Suppose that

$\sum\limits_{\lambda\vdash
n}c_{\lambda}s_{\lambda}=\sum\limits_{\alpha\models
n}d_{\alpha}F_{n,A(\alpha)},$

where
$A(\alpha)=(\alpha_{1},\alpha_{1}+\alpha_{2},\dots,\sum_{i=1}^{\ell(\alpha)-1}\alpha_{i})$.
Then we have $c_{(k,1^{n-k})}=d_{(k,1^{n-k})}$ for all $1\leq k\leq n$.

The original result from [ELW10] gives a description of the coefficients
$c_{\lambda}$ for a general $\lambda\vdash n$. However, since we only need the
case in the statement (i.e., when $\lambda$ is a hook), we refer interested
readers to [ELW10] for the general version, which is slightly more
complicated.

The following theorem is an immediate corollary of Theorem 3.13 and Theorem
3.14:

######  Theorem 3.15.

Let $(\bm{\nu},f)$ be an $r$-colored sequence of skew shapes of size $n$. Then
for any $1\leq k\leq n$

$[s_{(k,1^{n-k})}]\kappa_{\operatorname{LLT}}(\bm{\lambda})=\sum\limits_{\begin{subarray}{c}T\in\operatorname{SYT}(\bm{\lambda})\\\
\operatorname{Des}(T)=\\{k,k+1,\dots,n-1\\}\end{subarray}}\sum\mathcal{I}_{(\widehat{G_{\bm{\nu}}}^{E^{T}})_{f}}(q),$

where the second sum runs over all subsets $E^{T}\subseteq
E_{d}(G_{\bm{\nu}})$ for which $\widehat{G_{\bm{\nu}}}^{E^{T}}$ is
$f$-connected and $T(i)>T(j)$ whenever $(i,j)\in
E(\widehat{G_{\bm{\nu}}}^{E^{T}})$.

####  3.2.4. e-positivity

Let $(e_{\lambda})$ be the basis of elementary symmetric functions, i.e.
$e_{\lambda}:=\prod_{i=1}^{\ell(\lambda)}e_{\lambda_{i}}$, where
$e_{i}:=\sum_{j_{1}<\cdots<j_{i}}x_{j_{1}}\cdots x_{j_{i}}$ is the $i$-th
_elementary symmetric function_. $e$-positivity of a given symmetric function
$f$ is a stronger property than Schur-positivity and it suggests a specific
interpretation of the function $f$ in terms of the representation theory of
the symmetric group, and in algebro-geometric context. This observation
recently generated a lot of research in studying $e$-positive symmetric
functions, and after a series of conjectures [Ber17, AP18, GHQR19], it was
clear that $e$-positivity of a big class of symmetric functions would be a
consequence of $e$-positivity for _vertical-strip LLT polynomials_ after the
shift $q\to q+1$, i.e. for $\operatorname{LLT}(\bm{\nu})(q+1)$ where
$(\bm{\nu})_{i}=(1^{n_{i}+k_{i}})/(1^{k_{i}})$ for each $1\leq
i\leq\ell(\bm{\nu})$ and some nonnegative integers $n_{i},k_{i}$. An explicit
combinatorial formula for the coefficients of vertical-strip LLT polynomials
in the basis of elementary functions was independently conjectured in [GHQR19,
Ale21]222in fact, these interpretations are not identical, since the authors
use slightly different framework in their works, but it is possible to show
that they are equivalent and shortly afterwards the positivity (without
proving the combinatorial interpretation) was proved in [D’A20] and
subsequently [AS22] finalized the picture by proving the combinatorial
interpretation. In the following, we reformulate this combinatorial
interpretation in our current framework.

Let $\bm{\nu}$ be a tuple of vertical-strips and let $G=G_{\bm{\nu}}$ be the
associated LLT-graph. We recall that a vertex $v\in V(G)$ is associated with a
box $\square(v)\in\bm{\nu}$ and the vertices are naturally labeled by the
shifted contents of the corresponding boxes
$\tilde{c}(v):=\tilde{c}(\square(v))$. Fix $E\subset E_{d}(G)$ and define
$G^{\prime}=G^{E}$. Since $G^{\prime}$ is a directed graph (note that the
condition that $\bm{\nu}$ is a tuple of vertical strips implies that
$G^{\prime}$ has only edges of type I), some of the vertices of $G^{\prime}$
have only outgoing edges – such vertices are called _sources_. We define the
following equivalence relation on the set of vertices $V(G^{\prime})$: the
vertices $v\sim w$ are in the same equivalence class if the source $\theta(v)$
with the smallest label from which there exists a directed path to $v$ is the
same as the source $\theta(w)$ with the smallest label from which there exists
a directed path to $w$. The partition $\lambda(G^{\prime})$ is defined as the
partition whose parts are sizes of the equivalence classes in this relation.
See Fig. 3 for an example.

Figure 3. A tuple $\bm{\nu}$ of vertical strips and the associated LLT graph
$G_{\bm{\nu}}$. The labels of vertices are the shifted contents of the
corresponding boxes. Graph $G^{\prime}=G^{E}$ for
$E=\\{(0,3),(3,5),(4,5),(4,6)\\}$ is $f$-connected, where $f(i)=i$. In the
last picture, the displayed labels are the labels of the sources of
$G^{\prime}$ and the corresponding two equivalence classes $\\{0,3,5\\}$ and
$\\{4,6,7\\}$ are depicted by the whole and the empty vertices, respectively.

######  Theorem 3.16.

[AS22] Let $\bm{\nu}$ be a tuple of vertical-strips and let $G=G_{\bm{\nu}}$
be the associated LLT-graph. Then

(33) $\operatorname{LLT}(\bm{\nu})(q+1)=\sum_{E\subseteq
E_{d}(G)}q^{|E|}e_{\lambda(G^{E})}.$

In the following, we show that the vertical-strip LLT cumulants preserve
$e$-positivity, which refines Theorem 3.16, but, most importantly, shows that
$e$-positivity of vertical-strip LLT polynomials naturally decomposes into
$f$-connected components, each corresponding to the vertical-strip LLT
cumulant. In other terms, heuristically, the e-positivity of vertical-strip
LLT polynomials is “built” from e-positivity of LLT cumulants, which naturally
decompose LLT polynomials from the graph-coloring point of view.

######  Theorem 3.17.

Let $(\bm{\nu},f)$ be an $r$-colored tuple of vertical-strips and let
$G=G_{\bm{\nu}}$ be the associated LLT-graph. Then

(34)
$\kappa_{\operatorname{LLT}}(\bm{\nu},f)(q+1)=\sum\limits_{\begin{subarray}{c}E\subseteq
E_{d}\\\ G^{E}\text{
$f$-connected}\end{subarray}}q^{|E|+1-r}e_{\lambda(G^{E})}.$

###### Proof.

We recall that the $q$-partial cumulant of the family $(\bm{u})$ is defined by
the formula (2.3). One can invert this formula in order to express $u_{I}$ in
terms of the $q$-partial cumulants:

$u_{I}=\sum_{\pi\in\mathcal{P}(I)}(q-1)^{|I|-|\pi|}\prod_{B\in\pi}\kappa_{B}^{(q)}(\bm{u}).$

Applying this to our setting, we obtain that for any $r\geq 1$ and for any
$r$-colored tuple $(\bm{\nu},f)$, one has

$\operatorname{LLT}(\bm{\nu})(q+1)=\sum_{\pi\in\mathcal{P}([1..r])}q^{r-|\pi|}\prod_{B\in\pi}\kappa_{\operatorname{LLT}}((\bm{\nu},f)^{B},f|_{B})(q+1),$

where $f|_{B}$ is the $|B|$-coloring of $(\bm{\nu},f)^{B}$ obtained from $f$
by restricting it to the preimage of $B$, i.e., $f|_{B}\colon f^{-1}(B)\to B$.

We prove (34) by induction on $r$. For $r=1$, the LHS of (34) is equal to
$\operatorname{LLT}(\bm{\nu})(q+1)$, while the RHS of (34) coincides with the
RHS of (33), because every $1$-colored graph is trivially $f$-connected. Let
$(\bm{\nu},f)$ be an $r$-colored tuple of vertical-strips with $r>1$. Let
$G^{\prime}=G^{E}$ for some $E\subseteq E_{d}(G)$. Note that decomposing
$G^{\prime}$ into $f$-connected components, we find a set-partition
$\pi\in\mathcal{P}([1..r])$ such that each $f$-connected component has a
vertex set $V_{B}:=\\{v\in V(G^{\prime})\text{ colored by }b\in B\\}$ for some
$B\in\pi$. Therefore, we can rewrite (33) as follows

$\operatorname{LLT}(\bm{\nu})(q+1)=\sum_{\pi\in\mathcal{P}([1..r])}\prod_{B\in\pi}\bigg{(}\sum\limits_{\begin{subarray}{c}E_{B}\subseteq
E_{d}(G_{B})\\\ G_{B}^{E_{B}}\text{
$f|_{B}$-connected}\end{subarray}}q^{|E_{B}|}\bigg{)}e_{\lambda(\bigoplus_{B}G_{B}^{E_{B}})}.$

Notice also that
$e_{\lambda(\bigoplus_{B}G|_{V_{B}}^{E_{B}})}=\prod_{B\in\pi}e_{\lambda(G|_{V_{B}}^{E_{B}})}$,
which is immediate from the definition of $\lambda(G^{\prime})$. Indeed, the
whole equivalence class has to be contained in the connected component of $G$,
which is further contained in the $f$-connected component. Using the obvious
identity

$q^{r-|\pi|}=\prod_{B\in\pi}q^{|B|-1},$

we obtain

$\displaystyle\kappa_{\operatorname{LLT}}(\bm{\nu},f)(q+1)=\sum_{\pi\in\mathcal{P}([1..r])}q^{1-|\pi|}\prod_{B\in\pi}\bigg{(}\sum\limits_{\begin{subarray}{c}E_{B}\subseteq
E_{d}(G_{B})\colon\\\ G_{B}^{E_{B}}\text{ is
$f|_{B}$-connected}\end{subarray}}q^{|E_{B}|}\bigg{)}e_{\lambda(\bigoplus_{B}G_{B}^{E_{B}})}$
$\displaystyle-$
$\displaystyle\sum\limits_{\begin{subarray}{c}\pi\in\mathcal{P}([1..r])\\\
\pi\neq\\{[1..r]\\}\end{subarray}}q^{r-|\pi|}\prod_{B\in\pi}\kappa_{\operatorname{LLT}}((\bm{\nu},f)^{B},f|_{B})(q+1)=\sum\limits_{\begin{subarray}{c}E\subseteq
E_{d}\\\ G^{E}\text{
$f$-connected}\end{subarray}}q^{|E|+1-r}e_{\lambda(G^{E})},$

where the last equality follows from the inductive hypothesis, and the proof
is finished. ∎

##  4\. Concluding remarks and questions

We conclude by proving 2.7 for some special cases and stating some more
general open questions.

We start by showing that 2.7 holds true when $\ell(\bm{\nu})=2$.

######  Proposition 4.1.

Let $\bm{\nu}=((\lambda^{1}/\mu^{1},\lambda^{2}/\mu^{2}),f)$ be an $r$-colored
pair of skew Young diagrams. Then, for every partition $\lambda$ the
coefficient

$[s_{\lambda}]\kappa_{\operatorname{LLT}^{\operatorname{cospin}}}(\bm{\nu},f)\in\mathbb{Z}_{\geq
0}[q]$

is a polynomial in $q$ with nonnegative integer coefficients.

###### Proof.

We know that LLT polynomials are Schur positive, i.e

$\operatorname{LLT}^{\operatorname{cospin}}(\bm{\nu})(q)=\sum_{\lambda}c_{\lambda^{1}/\mu^{1},\lambda^{2}/\mu^{2}}^{\lambda}(q)s_{\lambda},$

where
$c_{\lambda^{1}/\mu^{1},\lambda^{2}/\mu^{2}}^{\lambda}(q)=\sum_{i=0}^{d_{\lambda^{1}/\mu^{1},\lambda^{2}/\mu^{2}}^{\lambda}}c_{\lambda^{1}/\mu^{1},\lambda^{2}/\mu^{2}}^{\lambda;i}q^{i}\in\mathbb{Z}_{\geq
0}[q]$ and we know that

$\operatorname{LLT}^{\operatorname{cospin}}(\lambda^{1}/\mu^{1})(q)\operatorname{LLT}^{\operatorname{cospin}}(\lambda^{2}/\mu^{2})(q)=\operatorname{LLT}^{\operatorname{cospin}}(\bm{\nu})(1).$

Therefore, the case of $2$-coloring gives us