Falling stars: a fall-decorated rational shuffle theorem

TL;DR

This paper introduces a rational analog of the fall Delta theorem and Delta square conjecture, defining a new dinv statistic on fall-decorated paths and proving related symmetric function formulas.

Contribution

It extends dinv statistics to fall-decorated paths and formulates a rational analog of key conjectures in algebraic combinatorics.

Findings

Defined a new dinv statistic on fall-decorated paths.

Proved a symmetric function formula for the q,t-generating function.

Established an analog formula for fall-decorated rectangular paths, conditional on a conjecture.

Abstract

In this paper, we formulate a rational analog of the fall Delta theorem and the Delta square conjecture. We find a new dinv statistic on fall-decorated paths on a $(m+k) \times (n+k)$ rectangle that simultaneously extends the previously known dinv statistics on decorated square objects and non-decorated rectangular objects. We prove a symmetric function formula for the $q,t$-generating function of fall-decorated rectangular Dyck paths as a skewing operator applied to $e_{m,n+km}$ and, conditionally on the rectangular paths conjecture, an analog formula for fall-decorated rectangular paths.

Equations

$$s_{(m-1{)}^k}^{\perp }{e}_{m,n+km}=\sum _{\pi \in \mathsf{LRD}(m+k,n+k{)}_{\ast k}}{q}^{\mathsf{dinv}(\pi )}{t}^{\mathsf{area}(\pi )}{x}^{\pi }.$$

$$s_{(m-1{)}^k}^{\perp }\frac{[m{]}_q}{[d{]}_q}{p}_{m,n+km}=\sum _{\pi \in \mathsf{LRP}(m+k,n+k{)}_{\ast k}}{q}^{\mathsf{dinv}(\pi )}{t}^{\mathsf{area}(\pi )}{x}^{\pi }.$$

$${\tilde{H}}_{\mu }:={\tilde{H}}_{\mu }[X]={\tilde{H}}_{\mu }[X;q,t]=\sum _{\lambda ⊢n}{\tilde{K}}_{\lambda \mu }(q,t)s_{\lambda }$$

$${\tilde{K}}_{\lambda \mu }:={\tilde{K}}_{\lambda \mu }(q,t)=K_{\lambda \mu }(q,1/t)t^{n(\mu )}$$

$$B_{\mu }:=B_{\mu }(q,t)=\sum _{c\in \mu }{q}^{a_{\mu }^{\prime }(c)}{t}^{l_{\mu }^{\prime }(c)},{\Pi }_{\mu }:={\Pi }_{\mu }(q,t)=\prod _{c\in \mu /(1)}(1-q^{a_{\mu }^{\prime }(c)}{t}^{l_{\mu }^{\prime }(c)}).$$

$$\nabla {\tilde{H}}_{\mu }=e_{|\mu |}[B_{\mu }]{\tilde{H}}_{\mu }.$$

$$\mathbf{\Pi }{\tilde{H}}_{\mu }={\Pi }_{\mu }{\tilde{H}}_{\mu }$$

$${\Delta }_f{\tilde{H}}_{\mu }=f[B_{\mu }]{\tilde{H}}_{\mu },{\Delta }_f^{\prime }{\tilde{H}}_{\mu }=f[B_{\mu }-1]{\tilde{H}}_{\mu }.$$

$$\Theta_{f}F\coloneqq\left\{\begin{array}[]{ll}0&\text{if }n\geq 1\text{ and }m=0\\ f\cdot F&\text{if }n=0\text{ and }m=0\\ \mathbf{\Pi}f\left[\frac{X}{M}\right]\mathbf{\Pi}^{-1}F&\text{otherwise}\end{array}\right.,$$

$${\Theta }_f{\Lambda }^{(n)}\subseteq {\Lambda }^{(n+k)}\text{ for }f\in {\Lambda }^{(k)}.$$

$$Q_{m,n}=\frac{1}{M}\left(Q_{c,d}{Q}_{a,b}-Q_{a,b}{Q}_{c,d}\right),$$

$$Q_{1,0}=D_0=\mathsf{id}-M{\Delta }_{e_1}\text{ and }{Q}_{0,1}=-\underline{e_1}$$

$$f=\sum _{\lambda ⊢d}{c}_{\lambda }(q,t){\left(\frac{qt}{qt-1}\right)}^{\ell (\lambda )}{h}_{\lambda }\left[\frac{1-qt}{qt}X\right].$$

$$F_{a,b}(f):=\sum _{\lambda ⊢d}{c}_{\lambda }(q,t)\prod _{i=1}^{\ell (\lambda )}{Q}_{{\lambda }_{i}a,{\lambda }_{i}b}(1).$$

$$e_{m,n}:=F_{a,b}(e_d),p_{m,n}:=F_{a,b}(p_d)$$

$$\left(-\frac{2}{7},-\frac{11}{7},\frac{1}{7},-\frac{8}{7},\frac{4}{7},-\frac{5}{7},0\right).$$

$${\mathbb{Z}}_{+}\cup {\overline{\mathbb{Z}}}_{+}=1<2<3<\cdots <\overline{1}<\overline{2}<\overline{3}<\dots .$$

$$(v_i,i){<}_{\text{lex}}(v_j,j){<}_{\text{lex}}(v_i+1,i).$$

$$\mathsf{tdinv}(\pi ,w):=\#\{i,j\in \mathcal{V}\mid w_i<w_j\text{ and }i\to j\}.$$

$$D_{+}$$

$$D_{-}$$

$$D_{+}^{\ast }$$

$$D_{-}^{\ast }$$

$$B$$

$$B^{\ast }$$

$$\mathsf{cdinv}(\pi ,\mathcal{D}):=\#D_{+}-\#D_{-}+\#D_{+}^{\ast }-\#D_{-}^{\ast }+\#B-\#B^{\ast }.$$

$$C_{+}$$

$$C_{-}$$

$$C^{\ast }$$

$$\mathsf{cdinv}(\pi ,\mathcal{D}):=\#C_{+}-\#C_{-}+\#C^{\ast }+\#B.$$

$$\#D_{+}-\#D_{-}+\#D_{+}^{\ast }-\#D_{-}^{\ast }-\#B^{\ast }=\#C_{+}-\#C_{-}+\#C^{\ast }.$$

$$\#C_{+}-\#C_{-}$$

$$\#C^{\ast }-\#E$$

$${\delta }_{[0,\frac{n}{m}+r_j-1)}(v_i-v_j)+{\delta }_{[\frac{n}{m}-1,0)}(v_i-v_j){\delta }_{r_j\ne 0}=\sum _{p=1}^{r_j}{\delta }_{[\frac{n}{m}+p-2,\frac{n}{m}+p-1)}(v_i-v_j)+{\delta }_{[0,\frac{n}{m}-1)}(v_i-v_j).$$

Full text

1. Introduction

In the 90’s, Garsia and Haiman set out to prove the Schur positivity of the (modified) Macdonald polynomials by showing them to be the bi-graded Frobenius characteristic of certain Garsia-Haiman modules [GarsiaHaiman1993GradedRepresentationModel]. Their prediction was confirmed in 2001, when Haiman used the algebraic geometry of the Hilbert scheme to prove that the dimension of their modules equals $n!$ [Haiman2001nFactorial], thus proving the $n!$ theorem. In the course of these developments, it became clear that there were remarkable connections to be found between Macdonald polynomial theory and representation theory of the symmetric group. For example, during their quest for Macdonald positivity, Garsia and Haiman introduced the $\mathfrak{S}_{n}$-module of diagonal harmonics, i.e. the coinvariants of the diagonal action of $\mathfrak{S}_{n}$ on polynomials in two sets of $n$ variables, and they conjectured that its Frobenius characteristic is given by $\nabla e_{n}$, where $\nabla$ is the nabla operator on symmetric functions introduced in [BergeronGarsiaHaimanTesler1999IdentitiesPositivityConjectures], which acts diagonally on Macdonald polynomials. Haiman proved this conjecture in 2002 [Haiman2002HilbertScheme].

The combinatorial side of things solidified when Haglund, Haiman, Loehr, Remmel, and Ulyanov then formulated the so called shuffle conjecture [HaglundHaimanLoehrRemmelUlyanov2005ShuffleConjecture], i.e. they predicted a combinatorial formula for $\nabla e_{n}$ in terms of labeled Dyck paths, which are lattice paths using North and East steps going from $(0,0)$ to $(n,n)$ and staying weakly above the line connecting these two points (called the main diagonal). Several years later, Haglund, Morse and Zabrocki conjectured a compositional refinement of the shuffle conjecture, which also specified all the points where the Dyck paths returns to main diagonal [HaglundMorseZabrocki2012CompositionalShuffleConjecture]. This was the statement later proved by Carlsson and Mellit in [CarlssonMellit2018ShuffleConjecture], implying the shuffle theorem.

Over the years, this subject has revealed itself to be extremely fruitful and to have striking connections to other fields of mathematics including elliptical Hall algebras, affine Hecke algebras, Springer fibers, the homology of torus knots and the shuffle algebra of symmetric functions.

In this paper, we add a few (conjectural) formulas to the substantial list of variants and generalizations inspired by the the success story of the shuffle theorem; that is, equations with a symmetric function related to Macdonald polynomials on one side and lattice paths combinatorics on the other. Furthermore, we support one of these conjectures by proving a non-trivial special case.

One of the earliest shuffle-like formulas was conjectured in 2007 Loehr and Warrington [LoehrWarrington2007Square]. They predicted an expression of $\nabla\omega(p_{n})$ in terms of square paths, i.e. lattice paths from $(0,0)$ to $(n,n)$ using only North and East steps and ending with an East step (without a the restriction of staying above the main diagonal). Their formula was proved by Sergel in [Sergel2017SquarePaths] to be a consequence of the shuffle theorem.

Next, Haglund, Remmel and Wilson formulated the Delta conjecture [HaglundRemmelWilson2018DeltaConjecture], a pair of conjectures for the symmetric function $\Delta^{\prime}_{e_{n-k-1}}e_{n}$ in terms of decorated Dyck paths, where $k$ decorations are placed on either rises or valleys of the path. The symmetric function operator $\Delta^{\prime}_{f}$ acts diagonally on the Macdonald polynomials and generalizes $\nabla$, in a sense. The rise version of the Delta conjecture was proved by D’Adderio and Mellit in [DAdderioMellit2022CompositionalDelta], using the compositional refinement in [DAdderioIraciVandenWyngaerd2021ThetaOperators]. A Delta square conjecture was stated in [DAdderioIraciVandenWyngaerd2019DeltaSquareConjecture] and is still open today; it extends (the rise version of) the Delta conjecture in the same fashion as the square paths theorem extends the shuffle theorem. The valley version also has similar extensions [QiuWilson2020ValleyExtendedDelta, IraciVandenWyngaerd2021ValleySquare], but it lacks a compositional version and it is still open.

Around the same time as the formulation of the Delta conjecture, the story has been extended to rectangular Dyck paths: paths from $(0,0)$ to $(m,n)$ staying above the main diagonal. In [BergeronGarsiaSergelLevenXin2016PlethysticOperators], building on the work in [GorskyNegut2015RefinedKnotInvariants], Bergeron, Garsia, Sergel, and Xin conjectured that a certain symmetric function related to the elliptic Hall algebra studied by Schiffmann and Vasserot [SchiffmannVasserot2011EllipticHallAlgebra] can be expressed in terms of rectangular Dyck paths. Their prediction was recently proved by Mellit [Mellit2021Rational].

A first attempt to unify these last two shuffle-like conjectures appears in [IraciPagariaPaoliniVandenWyngaerd2023], where the authors of the present paper and Vanden Wyngaerd propose a univariate rational rise Delta conjecture, in terms of Dyck paths on a rectangle that lie above a certain “broken” diagonal. In the present work, we perfect the attempt, giving a complete formulation in terms of fall-decorated Dyck paths instead, which we manage to prove exploiting the techniques introduced by Gillespie, Gorski, and Griffin in [GillespieGorskyGriffin2025].

Our main result is a rational shuffle theorem for rectangular paths, which generalizes both the rational shuffle theorem for Dyck paths in [Mellit2021Rational] (cf. [BlasiakHaimanMorsePunSeelinger2023ShuffleAnyLine]) and the rise Delta theorem in [HaglundRemmelWilson2018DeltaConjecture, DAdderioMellit2022CompositionalDelta].

Theorem 1.1** (Fall-decorated rational shuffle theorem).**

For any $m,n,k\in\mathbb{N}$ with $m>0$, we have

[TABLE]

The same approach works for paths above any “broken” line (see Definition 3.7 and Remark 3.8 for details), but we will restrict to paths from $(0,0)$ to $(m+k,n+k)$ for simplicity. Conditionally to [IraciPagariaPaoliniVandenWyngaerd2023]*Conjecture 4.2, we also prove the following.

Theorem 1.2** (Fall-decorated rectangular shuffle theorem).**

If [IraciPagariaPaoliniVandenWyngaerd2023]*Conjecture 4.2 holds, then for any $m,n,k\in\mathbb{N}$, and $d=\gcd(m,n)$, we have

[TABLE]

In particular, the identity holds if $d=1$.

The paper is structured as follows. In Section 2, we introduce the notation for the symmetric functions we use, and in Section 3, we do the same for the combinatorial objects.

Section 4 and Section 5 are set up to the proof of Theorem 1.1, which will be completed in Section 6. In Section 4, we adapt the main arguments in [GillespieGorskyGriffin2025] to our case. First, in Subsection 4.1, we show that applying $s^{\perp}_{(m-1)^{k}}$ to $e_{m,n+km}$ gives a signed sum of a certain subset of labeled rectangular paths of size $m\times(n+km)$. Then, in Subsection 4.2 we describe a bijection between a certain subset of labeled rectangular paths of size $m\times(n+km)$ and all fall-labeled rectangular paths of size $(m+k)\times(n+k)$ with $k$ decorated falls, without any restriction on being above the main diagonal. This bijection makes use of a new representation for decorated objects where decorated East steps are replaced by South steps; this representation preserves the area and the vertical distances of the vertical steps from the main diagonal. Afterwards, in Subsection 4.3 we exhibit a sign-reversing involution on our set with a unique fixed point for each element in $\mathsf{LRP}(m+k,n+k)_{\ast k}$. Finally, in Section 5 we show that the dinv of the fixed point is equal to the dinv of the corresponding element in $\mathsf{LRP}(m+k,n+k)_{\ast k}$, and in Section 6 we put together the pieces, completing the proof of Theorem 1.1.

We conclude the paper by highlighting some connections with the existing literature in Section 7, especially concerning the results about $D_{\alpha}$ operators by Blasiak et al. [BlasiakHaimanMorsePunSeelinger2023ShuffleAnyLine], and the Theta operators by D’Adderio et al. [DAdderioIraciVandenWyngaerd2021ThetaOperators].

2. Symmetric functions

For all the undefined notations and the unproven identities, we refer to [DAdderioIraciVandenWyngaerd2022TheBible]*Section 1, where definitions, proofs and/or references can be found. See also [IraciPagariaPaoliniVandenWyngaerd2023].

We denote by $\Lambda$ the graded algebra of symmetric functions with coefficients in $\mathbb{Q}(q,t)$, and by $\langle\,,\rangle$ the Hall scalar product on $\Lambda$, defined by declaring that the Schur functions form an orthonormal basis.

The standard bases of the symmetric functions that will appear in our calculations are the monomial $\{m_{\lambda}\}_{\lambda}$, complete $\{h_{\lambda}\}_{\lambda}$, elementary $\{e_{\lambda}\}_{\lambda}$, power $\{p_{\lambda}\}_{\lambda}$ and Schur $\{s_{\lambda}\}_{\lambda}$ bases.

For $f\in\Lambda$, we denote by $f^{\perp}$ the operator that is the adjoint of the product by $f$ with respect to the Hall scalar product, that is, for every $g,h\in\Lambda$, we have $\langle f^{\perp}g,h\rangle=\langle g,fh\rangle$.

For a partition $\mu\vdash n$, we denote by

[TABLE]

the (modified) Macdonald polynomials, where

[TABLE]

are the (modified) Kostka coefficients (see [Haglund2008Book]*Chapter 2 for more details).

Macdonald polynomials form a basis of the algebra of symmetric functions $\Lambda$. This is a modification of the basis introduced by Macdonald [Macdonald1995Book].

If we identify the partition $\mu$ with its Ferrer diagram, i.e. with the collection of cells $\{(i,j)\mid 1\leq i\leq\mu_{j},1\leq j\leq\ell(\mu)\}$, then for each cell $c\in\mu$ we refer to the arm, leg, co-arm and co-leg (denoted respectively by $a_{\mu}(c),l_{\mu}(c),a_{\mu}^{\prime}(c),l_{\mu}^{\prime}(c)$) as the number of cells in $\mu$ that are strictly to the right, below, to the left and above $c$ in $\mu$, respectively (see Figure 1).

Let $M\coloneqq(1-q)(1-t)$. For every partition $\mu$, we define the following constants:

[TABLE]

We will make extensive use of the plethystic notation (cf. [Haglund2008Book]*Chapter 1). We also need several linear operators on $\Lambda$.

Definition 2.1** ([BergeronGarsia1999ScienceFiction]*3.11).**

We define the linear operator $\nabla\colon\Lambda\rightarrow\Lambda$ on the eigenbasis of Macdonald polynomials as

[TABLE]

Definition 2.2**.**

We define the linear operator $\mathbf{\Pi}\colon\Lambda\rightarrow\Lambda$ on the eigenbasis of Macdonald polynomials as

[TABLE]

where we conventionally set $\Pi_{\varnothing}\coloneqq 1$.

Definition 2.3**.**

For $f\in\Lambda$, we define the linear operators $\Delta_{f},\Delta^{\prime}_{f}\colon\Lambda\rightarrow\Lambda$ on the eigenbasis of Macdonald polynomials as

[TABLE]

Observe that on the vector space of homogeneous symmetric functions of degree $n$, denoted by $\Lambda^{(n)}$, the operator $\nabla$ equals $\Delta_{e_{n}}$.

Definition 2.4** ([DAdderioIraciVandenWyngaerd2021ThetaOperators]*(28)).**

For any symmetric function $f\in\Lambda^{(n)}$ we define the Theta operators on $\Lambda$ in the following way: for every $F\in\Lambda^{(m)}$ we set

[TABLE]

and we extend by linearity the definition to any $f,F\in\Lambda$.

It is clear that $\Theta_{f}$ is linear. In addition, if $f$ is homogeneous of degree $k$, then so is $\Theta_{f}$:

[TABLE]

Finally, we need to refer to [BergeronGarsiaSergelLevenXin2015ShuffleConjectures]*Algorithm 4.1 (see also [BergeronGarsiaSergelLevenXin2016PlethysticOperators]*Definition 1.1, Theorem 2.5).

Definition 2.5**.**

Let $m,n>0$. Let $a,b,c,d\in\mathbb{N}$ such that $a+c=m$, $b+d=n$, $ad-bc=\gcd(m,n)$. We recursively define $Q_{m,n}$ as an operator on $\Lambda$ by

[TABLE]

with base cases

[TABLE]

(where $\underline{f}$ is the multiplication by $f$).

Definition 2.6**.**

For a coprime pair $(a,b)$ and $f\in\Lambda^{(d)}$, we define $F_{a,b}(f)$ as follows. Let

[TABLE]

Then, we define

[TABLE]

In the literature, $F_{a,b}(f)$ is often written as $f[-MX^{a,b}]$. We use the shorthands

[TABLE]

where $m=ad,n=bd$, and $\gcd(a,b)=1$. Beware: $e_{4,2}=F_{2,1}(e_{2})$, but $e_{42}=e_{4}e_{2}$.

3. Combinatorial definitions

The objects we are concerned with are rectangular Dyck paths and rectangular paths. The following definitions first appeared in [BergeronGarsiaSergelLevenXin2016PlethysticOperators] for rectangular Dyck paths and in [IraciPagariaPaoliniVandenWyngaerd2023] for rectangular paths.

3.1. Rectangular paths

Definition 3.1**.**

A rectangular path of size $m\times n$ is a lattice path composed of unit vertical and horizontal steps, going from $(0,0)$ to $(m,n)$, and ending with a horizontal step. A rectangular Dyck path is a rectangular path that lies weakly above the diagonal $my=nx$ (called main diagonal).

We denote the sets of rectangular paths of size $m\times n$ as $\mathsf{RP}(m,n)$.

Given a rectangular path $\pi$, denote by $\mathcal{H}=\mathcal{H}(\pi)$ (resp. $\mathcal{V}=\mathcal{V}(\pi)$) the set of horizontal (resp. vertical) steps of $\pi$. The set $\mathcal{H}\sqcup\mathcal{V}$ is totally ordered by traversing the path from $(0,0)$ to $(m,n)$: we write $i<j$ if step $i$ precedes step $j$. If $i$ is a step of $\pi$ and $p\in\mathbb{Z}$ is an integer, we write $i+p$ to denote the step that lies $p$ positions away from $i$ along $\pi$; thus $i+1$ is the step immediately after $i$ (if it exists).

Definition 3.2**.**

For a $m\times n$ rectangular path $\pi$ and a step $i\in\mathcal{H}$ (resp. $i\in\mathcal{V}$), denote by $v_{i}=v_{i}(\pi)$ the (signed) vertical distance between the right-most (resp. bottom-most) point of $i$ and the main diagonal; the sign is positive if the endpoint is above the main diagonal. Define the vertical area word of the path as the sequence $(v_{i})_{i\in\mathcal{H}}$, i.e., the sequence of heights of the horizontal steps of the path. Set $s\coloneqq-\min\{v_{i}\mid i\in\mathcal{H}\}$, which we call the shift of the path. Note that $s=0$ if $\pi$ is a rectangular Dyck path, and $s>0$ otherwise.

Definition 3.3**.**

The diagonal $m(y+s)=nx$ is called the base diagonal. It is the lowest diagonal that intersects the path.

Definition 3.4**.**

The area of a rectangular path $\pi$ is $\mathsf{area}(\pi)\coloneqq\sum_{i\in\mathcal{H}}\lfloor v_{i}+s\rfloor$. This is the number of whole squares that lie entirely between the path $\pi$ and its base diagonal (including the squares below the line $y=0$ but excluding the squares to the right of the line $x=m$).

Remark 3.5*.*

This definition differs from [IraciPagariaPaoliniVandenWyngaerd2023]*Definition 3.4 as we count the squares that are vertically below the path instead of the squares that are horizontally right of the path. This is irrelevant for now, as the two triangles outside of the $m\times n$ rectangle we are counting in the two cases are congruent and translated by an integer vector. However, it will become important when we define the area of decorated paths.

For example, the path in Figure 2 has vertical area word

[TABLE]

Thus, its shift is $\frac{11}{7}$ and its area is $1+0+1+0+2+0+1=5$.

3.2. Decorated rectangular paths

In a similar fashion as the rise version of the Delta conjecture [HaglundRemmelWilson2018DeltaConjecture] (which is now a theorem [DAdderioMellit2022CompositionalDelta, BlasiakHaimanMorsePunSeelinger2023ProofExtendedDelta]), we introduce the concept of decorated falls for rectangular paths.

Definition 3.6**.**

The falls of a rectangular path are the horizontal steps that are immediately followed by another horizontal step. A fall-decorated rectangular path is a pair $(\pi,\mathcal{D})$ where $\pi$ is a rectangular path and $\mathcal{D}$ is a subset of its falls. Normally, we will simply write decorated in place of fall-decorated. For a decorated rectangular path, we denote by $\mathcal{H}$ the set of non-decorated horizontal steps, so that the set of all steps of $\pi$ is the disjoint union $\mathcal{H}\sqcup\mathcal{V}\sqcup\mathcal{D}$ (totally ordered by traversing the path).

Definition 3.7**.**

For a decorated rectangular path of size $(m+k)\times(n+k)$ with $k$ decorated falls, we define the broken diagonal to be the broken line segment from $(0,0)$ to $(m+k,n+k)$ that proceeds with slope $\frac{n}{m}$ in columns containing non-decorated horizontal steps, and with slope $1$ in columns containing decorated falls.

This definition is analog to [IraciPagariaPaoliniVandenWyngaerd2023]*Definition 3.6 when reflecting the path with respect to the antidiagonal. Note that, if the path has no decorated falls, then the broken diagonal coincides with the main diagonal.

Since the broken diagonal never proceeds horizontally or vertically, it intersects any horizontal or vertical segment in at most one point. We say that a point $(x,y)$ with $0\leq x\leq m$ diagonally projects onto a horizontal or vertical step if the vertical translation of the broken diagonal passing through $(x,y)$ intersects the step.

Remark 3.8*.*

The definition of broken diagonal extends to any slope, as follows. Fix any positive reals $a$ and $b$, and let $s=b/a$. Construct any broken line segment from $(0,-\{b\})$ to $(a+k,\lfloor b\rfloor+k)$ that proceeds with slope $1$ in $k$ of the columns and with slope $s$ in the remaining columns; here, $\{b\}\coloneqq b-\lfloor b\rfloor$ denotes the fractional part. Consider all fall-decorated rectangular paths from $(0,0)$ to $(\lfloor a\rfloor+k,\lfloor b\rfloor+k)$ that lie above this line and having $k$ decorated falls exactly in the chosen columns of slope $1$. All results in this paper extend to these paths as well, but we will only treat the case where $a,b\in\mathbb{N}$ for simplicity.

Remark 3.9*.*

If $m=n$, that is, the path is a square path, then there is a bijection between falls and rises (i.e. vertical steps preceded by vertical steps). This is not the case for rectangular paths. However, the reflection with respect to the antidiagonal defines a bijection between $(m+k)\times(n+k)$ rectangular Dyck paths with $k$ decorated falls and $(n+k)\times(m+k)$ rectangular Dyck paths with $k$ decorated rises, as in [IraciPagariaPaoliniVandenWyngaerd2023]*Definition 3.5. This bijection will come into play in Section 7.2.

Definition 3.10**.**

We define a decorated rectangular Dyck path to be a decorated rectangular path that lies weakly above the broken diagonal.

See Figure 3 (left) for an example of such a path. In our figures, we use a $\ast$ to mark the decorated falls.

The definitions of vertical distance and vertical area word extend to decorated paths as well, using the broken diagonal in place of the main diagonal. The definition of area also extends to decorated paths, where the sum over $i\in\mathcal{H}$ excludes the decorated steps. For example, the area of the path in Figure 3 is equal to $3$.

An alternative way to draw a $(m+k)\times(n+k)$ rectangular path $\pi$ with $k$ decorated falls is achieved by replacing decorated horizontal steps $i\in\mathcal{D}$ by decorated South steps. The result is a lattice path from $(0,0)$ to $(m,n)$ with $m$ East steps (indexed by $\mathcal{H}$), $n+k$ North steps (indexed by $\mathcal{V}$), and $k$ decorated South steps (indexed by $\mathcal{D}$), where a sequence of consecutive South steps must be followed by an East step; see Figure 3 (right). We call this alternative representation the ENS representation of $\pi$. In this framework, the extension suggested in Remark 3.8 is even more apparent.

Conveniently, the vertical distance $v_{i}$ of a step from the broken diagonal (in the usual representation) coincides with the vertical distance of the corresponding step from the main diagonal in the ENS representation. Also, the area of a path $\pi$ is the number of whole squares between $\pi$ and its base diagonal in the ENS representation.

3.3. Labeled paths

Finally, we need to introduce labeled objects.

Definition 3.11**.**

A labeling of a decorated rectangular path is an assignment of a positive integer label to each vertical step of the path, such that consecutive vertical steps are assigned strictly increasing labels. A labeled decorated rectangular path is a triple $(\pi,\mathcal{D},w)$ where $(\pi,\mathcal{D})$ is a decorated rectangular path and $w$ is a labeling.

Remark 3.12*.*

With an abuse of notation, we will sometimes write $\pi$ to mean $(\pi,\mathcal{D},w)$.

We say that a labeling is standard if the set of labels is $[n+k]\coloneqq\{1,\dots,n+k\}$, where $n+k$ is the height of the path, and we denote by $w_{i}$ the label assigned to the vertical step $i\in\mathcal{V}$.

We denote the sets of labeled rectangular paths and labeled rectangular Dyck paths of size $m\times n$ by $\mathsf{LRP}(m,n)$ and $\mathsf{LRD}(m,n)$ respectively, and the sets of labeled fall-decorated rectangular paths and labeled fall-decorated rectangular Dyck paths of size $(m+k)\times(n+k)$ with $k$ decorations by $\mathsf{LRP}(m+k,n+k)_{\ast k}$ and $\mathsf{LRD}(m+k,n+k)_{\ast k}$, respectively.

Definition 3.13**.**

Given a labeled decorated rectangular path $(\pi,\mathcal{D},w)$, let $x^{w}=\prod_{i\in\mathcal{V}}x_{w_{i}}$. We sometimes write $x^{\pi}$ in place of $x^{w}$.

It will be useful later to consider labelings in the alphabet

[TABLE]

In this case, we set $x_{\overline{i}}=y_{i}$ in the plethystic alphabet $X+Y$.

We now extend the definition of dinv given in [IraciPagariaPaoliniVandenWyngaerd2023] (cfr. [BergeronGarsiaSergelLevenXin2016PlethysticOperators, HicksSergel2015SimplerFormulaNumber, Mellit2021Rational]) to any fall-decorated rectangular path.

Definition 3.14**.**

Let $\pi$ be a $(m+k)\times(n+k)$ (decorated) rectangular path, and let $i,j\in\mathcal{V}$. We say that $i$ attacks $j$ in $\pi$ (or $(i,j)$ is an attack relation for $\pi$), and write $i\to j$, if

[TABLE]

At this point, we can define the temporary dinv of a path.

Definition 3.15**.**

We define the temporary dinv of a labeled (decorated) rectangular path $(\pi,w)$ as

[TABLE]

Geometrically, the temporary dinv counts all pairs of vertical steps $(i,j)$ such that $w_{i}<w_{j}$ and the bottom-most endpoint of step $j$ diagonally projects onto step $i$, where we include the bottom-most endpoint of step $i$ if $i<j$ and the top-most endpoint of step $i$ if $i>j$.

In order to complete the definition of dinv, we need to introduce a correction term, which is the cdinv of the path. This is a generalization of the cdinv defined in [HicksSergel2015SimplerFormulaNumber] for Dyck paths and later in [IraciPagariaPaoliniVandenWyngaerd2023] for rectangular paths.

First, we introduce some notation.

Definition 3.16**.**

Let $(\pi,\mathcal{D})$ be a $(m+k)\times(n+k)$ fall-decorated rectangular path with $k$ decorated falls. For $j\in\mathcal{H}$, we define $r_{j}$ as the number of decorated falls immediately preceding the horizontal step $j$.

We are now ready to define the dinv correction (or cdinv). We are giving two different definitions, which we prove to be equivalent in Lemma 3.19.

Definition 3.17**.**

Let $(\pi,\mathcal{D})$ be a fall-decorated rectangular path, and let

[TABLE]

We define the dinv correction of $(\pi,\mathcal{D})$ as

[TABLE]

The set $D_{+}$ in Definition 3.17 consists of all pairs of vertical and horizontal steps $(i,j)$ such that the vertical step comes after the horizontal step and both endpoints of the horizontal step diagonally project onto the vertical step (top-most endpoint included, bottom-most endpoint excluded); this can only happen for $n<m$ and if step $j$ is not decorated.

The set $D_{-}$ consists of all pairs of vertical and horizontal steps $(i,j)$ such that the vertical step is above the horizontal step and both endpoints of the vertical step diagonally project onto the horizontal step (right-most endpoint included, left-most endpoint excluded); this can only happen for $n>m$ and if step $j$ is not decorated.

The set $D_{+}^{\ast}$ consists of all pairs of horizontal steps $i<j$ such that step $i$ is decorated and both endpoints of step $j$ diagonally project onto step $i$ (right-most endpoint included, left-most endpoint excluded); this can only happen for $n<m$ and if step $j$ is not decorated.

The set $D_{-}^{\ast}$ consists of all pairs of horizontal steps $i<j$ such that step $i$ is decorated and both endpoints of step $i$ diagonally project onto step $j$ (right-most endpoint excluded, left-most endpoint included); this can only happen for $n>m$ and if step $j$ is not decorated.

See Figure 5 (cf. Figure 6) for an illustration of the definitions of $D_{+}$, $D_{-}$, $D_{+}^{\ast}$, and $D_{-}^{\ast}$.

It is convenient to give an alternative, more compact, definition of the dinv correction.

Definition 3.18**.**

Let $(\pi,\mathcal{D})$ be a fall-decorated rectangular path, and let

[TABLE]

where $v_{j}^{+}\coloneqq v_{j}+\delta_{j\in\mathcal{D}}+\textstyle\frac{n}{m}\delta_{j\in\mathcal{H}}$ is the vertical distance between the left-most point of the horizontal step $j$ and the broken diagonal, which depends on whether or not $j$ is decorated. We define the dinv correction of $(\pi,\mathcal{D})$ as

[TABLE]

The set $C_{+}$ in Definition 3.18 contains the same pairs as the set $D_{+}$ in Definition 3.17, with the additional restriction that the horizontal step is not immediately preceded by a decorated fall.

The set $C_{-}$ contains the same pairs as the set $D_{-}$ in Definition 3.17, with the relaxed condition that the endpoints of the vertical step diagonally project onto the horizontal step or any of the decorated falls immediately preceding the horizontal step; the horizontal step is required to be non-decorated.

The set $C^{\ast}$ consists of all pairs of non-consecutive horizontal steps $i<j-1$ such that step $i$ is decorated and the left-most endpoint of step $j$ diagonally projects onto step $i$ (right-most endpoint included, left-most endpoint excluded). In the definition, it is important to specify $i<j-1$ (and not simply $i<j$) because otherwise $C^{\ast}$ would also contain all pairs of consecutive decorated falls, which we want to exclude.

See Figure 7 (cf. Figure 8) for an illustration of the definitions of $C_{+}$, $C_{-}$, and $C^{\ast}$.

Lemma 3.19**.**

For each path $(\pi,\mathcal{D})\in\mathsf{RP}(m+k,n+k)_{*k}$, we have

[TABLE]

In particular, Definitions 3.17 and 3.18 are equivalent.

Proof.

Let $E=\left\{(i,j)\in\mathcal{V}\times\mathcal{D}\mid j<i\text{ and }v_{i}<v_{j}\leq v_{i}+1\right\}$ be the set of pairs of vertical and horizontal steps $(i,j)$ such that the horizontal step is decorated, the vertical step comes after the horizontal step, and the left-most endpoint of the horizontal step diagonally projects onto the vertical step (top-most endpoint excluded, bottom-most included). The statement is implied by the following two equalities:

[TABLE]

For the first equality, consider a non-decorated horizontal step $j\in\mathcal{H}$ and a vertical step $i\in\mathcal{V}$ such that $j<i$. We have

[TABLE]

The first summand on the left-hand side determines whether the pair $(i,j)$ belongs to $C_{-}$, and the second determines whether it belongs to $D_{+}\setminus C_{+}$. The first $r_{j}$ summands on the right-hand side determine whether the pairs $(i,j-p)$ belong to $E$ (for $p=1,\dots,r_{j}$) and the last one whether $(i,j)$ belongs to $D_{-}$. Taking the sum over all pairs $(i,j)$, we obtain the first claimed equality.

Regarding the second equality, notice that

[TABLE]

and

[TABLE]

The claimed identity reduces to

[TABLE]

Fix a decorated horizontal step $j$ and look at all contributions in the above identity from pairs involving $j$ and horizontal or vertical steps to the right of $j$. Each time the path crosses the diagonal at height $v_{j}$, we have a contribution to the left-hand side with a sign. All such contributions cancel pairwise if $v_{j}>0$; otherwise, only one negative contribution survives and is counted by $B^{*}$. ∎

Definition 3.20**.**

We define the dinv of a labeled rectangular path $(\pi,w)$ as

[TABLE]

Note that the cdinv only depends on the path $\pi$, whereas the tdinv also depends on the labels $w$.

Remark 3.21*.*

When $k=0$ (i.e. there are no decorated falls), the sets $C_{+}$ and $C_{-}$ of Definition 3.18 reduce to the sets in [HicksSergel2015SimplerFormulaNumber]*Theorem 2 and Figure 3, which cannot be simultaneously non-empty, and $C_{*}$ is empty. Also notice that, when $m=n$ (i.e. the path is square), the sets $D_{+},D_{-},D_{+}^{\ast}$, and $D_{-}^{\ast}$ of Definition 3.17 are empty, so the cdinv reduces to $\#B-\#B^{\ast}$. In particular, our dinv statistic extends both the dinv of rectangular (Dyck) paths as defined in [HicksSergel2015SimplerFormulaNumber] and [IraciPagariaPaoliniVandenWyngaerd2023], and the dinv for rise-decorated Dyck paths of the rise Delta theorem in [HaglundRemmelWilson2018DeltaConjecture, DAdderioMellit2022CompositionalDelta].

4. Combinatorial arguments

4.1. Combinatorial interpretation of the skewing operator

Recall that

[TABLE]

for any symmetric functions $f,g\in\Lambda$, as the scalar product with any symmetric function in the alphabet $X$ is the same.

Since the homogeneous and the monomial symmetric functions are dual to each other, for any weak composition $\alpha$, knowing that

[TABLE]

is a symmetric function (as it is a positive sum of LLT polynomials), we can interpret the skewing operator $h_{\alpha}^{\perp}$ as extracting the coefficient of $y^{\alpha}$ in the expression above, where the labels of the vertical steps are in the alphabet $\mathbb{Z}_{+}\cup\overline{\mathbb{Z}}_{+}$ (and the labels in $\mathbb{Z}_{+}$ are smaller than the labels in $\overline{\mathbb{Z}}_{+}$), and then restrict to the alphabet $X$ (or equivalently, set $Y=0$). The same holds for $\mathsf{LRD}(m,n)$. This motivates the following definition.

Definition 4.1**.**

For $m,n,r\in\mathbb{N}$ and $\alpha\vDash r$ weak composition, we define the sets $\mathsf{LRD}(m,n)^{\alpha}$ and $\mathsf{LRP}(m,n)^{\alpha}$ to be the sets of rectangular (Dyck) paths of size $m\times n$ with labels in the alphabet $\mathbb{Z}_{+}\cup\overline{\mathbb{Z}}_{+}$ such that there are exactly $\alpha_{i}$ vertical steps with label $\overline{i}$ for all $i$. We refer to the labels in $\mathbb{Z}_{+}$ as small and to the labels in $\overline{\mathbb{Z}}_{+}$ as big. For $\pi\in\mathsf{LRP}(m,n)^{\alpha}$, we denote by $\mathcal{V}$ the set of vertical steps with small labels and by $\mathcal{B}$ the set of vertical steps with big labels, so that the set of all steps of $\pi$ is the disjoint union $\mathcal{H}\sqcup\mathcal{V}\sqcup\mathcal{B}$ (totally ordered by traversing the path).

As before, set $x^{\pi}=x^{w}=\prod_{i\in\mathcal{V}}x_{w_{i}}$; note that this product disregards the big labels. From the previous discussion, we deduce the following.

Proposition 4.2**.**

For any $m,n,r\in\mathbb{N}$ and $\alpha\vDash r$, we have

[TABLE]

and the same holds if we replace $\mathsf{LRP}(m,n)$ with $\mathsf{LRD}(m,n)$.

We now want to apply the skewing operator $s_{(m-1)^{k}}^{\perp}$ to $e_{m,n+km}$. In order to do so, we need the homogeneous expansion of $s_{(m-1)^{k}}$. Let us recall the definition of allowable composition.

Definition 4.3** ([GillespieGorskyGriffin2025]*Definition 4.22).**

Let $\alpha\vDash k$ be a weak composition with $\ell(\alpha)\leq k$. We say that $\alpha$ is allowable if $\alpha_{1}>0$, and each $\alpha_{i}>0$ is followed by exactly $\alpha_{i}-1$ zeros within the first $k$ entries (i.e. $\alpha_{i}+i-1\leq k$ for all $i$, and $\alpha_{i+1}=\dots=\alpha_{i+\alpha_{i}-1}=0$ and $\alpha_{i+\alpha_{i}}\neq 0$ unless $i+\alpha_{i}-1=k$).

We also define the sign of an allowable composition as

[TABLE]

Using the Jacobi-Trudi formula, we get the following.

Proposition 4.4** ([GillespieGorskyGriffin2025]*Equation (10)).**

For $m,k\in\mathbb{N}$, we have

[TABLE]

where $\tilde{\alpha}_{i}=m-\alpha_{i}$, and $\textnormal{sgn}(\alpha)=\#\{1\leq i\leq k\mid\alpha_{i}=0\}$ (recall that $h_{a}=0$ for all $a<0$).

4.2. Bijection with rectangular Dyck paths

Let $m,n,k\in\mathbb{N}$ and $\alpha\vDash k$ allowable. Let $\tilde{\alpha}\vDash k(m-1)$ be defined as $\tilde{\alpha}_{i}=m-\alpha_{i}$. In order to compute $s_{(m-1)^{k}}^{\perp}e_{m,n+km}$ in terms of our objects, we need a bijection between paths in $\mathsf{LRP}(m,n+km)^{\tilde{\alpha}}$ and a subset of paths in $\mathsf{LRP}(m+k,n+k)_{\ast k}$ whose labels depend on $\alpha$ (see Proposition 4.9). This map is an adaptation of [GillespieGorskyGriffin2025]*Definition 4.6. Note that both sets are empty if $\alpha_{i}>m$ for some $i$.

From now on, to improve readability, we will use $\tilde{\pi}$ to denote paths of size $m\times(n+km)$ and reserve $\pi$ for (decorated) paths of size $(m+k)\times(n+k)$.

Definition 4.5**.**

Let $\tilde{\pi}\in\mathsf{LRP}(m,n+km)^{\tilde{\alpha}}$. We define $\psi_{0}(\tilde{\pi})\in\mathsf{LRP}(m+k,n+k)_{\ast k}$ as the path obtained from $\tilde{\pi}$ by the following procedure:

1. (1)

delete all vertical steps $i\in\mathcal{B}$; 2. (2)

if immediately before the horizontal step $j\in\mathcal{H}$ there were $a_{j}$ such steps, replace them with $k-a_{j}$ decorated falls; 3. (3)

keep everything else in place, including the remaining labels.

Remark 4.6*.*

Via this map, we lose the information given by the big labels. However, since the labels in each column are a subset of $[\overline{k}]$, and they get replaced by a streak of decorated falls of complementary size, we can recover the information by labeling the decorated falls of $\psi_{0}(\tilde{\pi})$ with the complementary subset of $[\overline{k}]$.

This leads to the following definition.

Definition 4.7**.**

Let $\pi\in\mathsf{LRP}(m+k,n+k)_{\ast k}$. We define a fall-labeling of $\pi$ to be a function $\overline{w}\colon\mathcal{D}\to[\overline{k}]\coloneqq\{\overline{1},\dots,\overline{k}\}$ such that, if $i,i+1\in\mathcal{D}$, then $\overline{w}_{i}<\overline{w}_{i+1}$.

In other words, we assign a label from $\overline{1}$ to $\overline{k}$ to each decorated fall of the path, in such a way that the labels are strictly increasing from left to right. The content of a fall-labeling is the weak composition $\alpha\vDash k$ where

[TABLE]

that is, the number of times the label $i$ appears on a fall.

Definition 4.8**.**

Given a fall-labeling $\overline{w}$ of $\pi$, we define the star word of $\pi$ as the sequence $w^{\ast}(\pi,\overline{w})$ of labels of the decorated falls, read from the greatest to the smallest vertical distance, and from top to bottom in case of a tie. We say that a fall-labeling is allowable if $w^{\ast}(\pi,\overline{w})$ has allowable content.

Let $\mathsf{FL}(\pi)$ denote the set of allowable fall-labelings of $\pi$, and $\mathsf{FL}(\pi,\alpha)$ the subset of fall-labelings with content $\alpha$. The following proposition is immediate.

Proposition 4.9**.**

The map $\psi_{0}$ lifts to a bijection $\psi$ between $\mathsf{LRP}(m,n+km)^{\tilde{\alpha}}$ and the set

[TABLE]

where $\overline{w}$ is obtained as in Remark 4.6.

The definitions of $\mathsf{dinv}$, $\mathsf{tdinv}$, and all the other statistics extend to objects with a fall-labeling by disregarding it.

To better visualize the bijection $\psi$, we introduce an ENS representation for paths in $\mathsf{LRP}(m,n+km)^{\tilde{\alpha}}$ as follows: before every horizontal step $j$, insert $k$ South steps, where the first $a_{j}$ are non-decorated and the last $k-a_{j}$ are decorated (here, $a_{j}$ is as in Definition 4.5). The result is a lattice path from $(0,0)$ to $(m,n)$ with $m$ East steps (indexed by $\mathcal{H}$), $n+km$ North steps ($n+k$ of them being indexed by $\mathcal{V}$, and $k(m-1)$ being indexed by $\mathcal{B}$), and $km$ South steps. Denote by $\mathcal{D}$ the set of decorated South steps and by $\mathcal{S}$ the set of non-decorated South steps (naturally in bijection with $\mathcal{B}$); we have $\#\mathcal{D}=k$ and $\#\mathcal{S}=\#\mathcal{B}=k(m-1)$. See Figure 9 (top right).

When passing to the ENS representation, the vertical distance between the left-most endpoint of horizontal steps and the main diagonal is reduced by $k$; the vertical distances are preserved for all other endpoints. As illustrated in Figure 9, the map $\psi$ operates on the ENS representation simply by pruning away the vertical steps in $\mathcal{B}\sqcup\mathcal{S}$. The remaining steps $i\in\mathcal{H}\sqcup\mathcal{V}\sqcup\mathcal{D}$ are identified with the corresponding steps in $\psi(\tilde{\pi})$; when working with both $\tilde{\pi}$ and $(\pi,\overline{w})=\psi(\tilde{\pi})$, we will use the sets $\mathcal{H},\mathcal{V},\mathcal{D}$ to denote steps from both paths.

From the previous description of $\psi$, the following lemma is immediate.

Lemma 4.10**.**

Let $\tilde{\pi}\in\mathsf{LRP}(m,n+km)^{\tilde{\alpha}}$. For any step $i\in\mathcal{H}\sqcup\mathcal{V}$ (horizontal, or vertical with a small label), its vertical distance is preserved by $\psi$:

[TABLE]

In particular, we have $\mathsf{area}(\tilde{\pi})=\mathsf{area}(\psi(\tilde{\pi}))$ and $\mathsf{shift}(\tilde{\pi})=\mathsf{shift}(\psi(\tilde{\pi}))$. Therefore, $\psi$ restricts to a bijection between $\mathsf{LRD}(m,n+km)^{\tilde{\alpha}}$ and the set

[TABLE]

Remark 4.11*.*

Given $i,j\in\mathcal{V}$, we have $i\rightarrow j$ in $\tilde{\pi}$ if and only if $i\rightarrow j$ in $(\pi,\overline{w})=\psi(\tilde{\pi})$. Therefore, the pair $(i,j)$ contributes to $\mathsf{tdinv}(\tilde{\pi})$ if and only if it contributes to $\mathsf{tdinv}(\pi)$. The remaining contributions to $\mathsf{tdinv}(\tilde{\pi})$ are due to attacking pairs $(i,j)\in(\mathcal{V}\sqcup\mathcal{B})\times\mathcal{B}$. We call $\mathsf{falldinv}(\pi,\overline{w})$ the number of such contributions, so that

[TABLE]

4.3. Sign-reversing involution

Let $\pi\in\mathsf{LRP}(m+k,n+k)_{\ast k}$, and for $\overline{w}\in\mathsf{FL}(\pi)$, let $\textnormal{sgn}(\overline{w})$ to be the sign of its content. We aim to define a sign-reversing involution on $\mathsf{FL}(\pi)$ such that $\overline{w}$ is a fixed point if and only if $w^{\ast}(\pi,\overline{w})=\overline{1}\,\overline{2}\cdots\overline{k}$.

Let us recall the sign-reversing involution $\varphi$ on words with allowable content from [GillespieGorskyGriffin2025].

Definition 4.12** ([GillespieGorskyGriffin2025]*Definition 4.28).**

Let $a$ be any word of length $k$ with allowable content. Let $p$ be the largest entry in $a$ such that

• •

there is only one $p$, and

• •

if $q$ is the largest entry in $a$ that is less than $p$, then $p$ is to the left of every $q$.

Notice that such an $p$ might not exist. Let $r$ be the largest repeated entry in $a$, and let $s$ be the smallest entry larger than $m$ in $a$, or $s=k+1$ if $r$ is already the largest. Note that $p\neq r$. Then we define $\varphi(a)$ as the word obtained from $a$ by applying the following procedure:

• Case 1.

If $p>r$ or $r$ does not exist, replace $p$ with an occurrence of $q$.

• Case 2.

If $p<k$ or $p$ does not exist, replace the first occurrence of $r$ with $s-1$.

• Case 3.

If neither $p$ nor $r$ exist, do nothing.

We have the following result.

Theorem 4.13** ([GillespieGorskyGriffin2025]*Theorem 4.30).**

The map $\varphi$ is a sign-reversing involution on words with allowable content, preserves tied inversions (i.e. if $i<j$, then $a_{i}\geq a_{j}\iff\varphi(a)_{i}\geq\varphi(a)_{j}$), and its unique fixed point is $12\cdots k$, which has positive sign.

We can lift $\varphi$ to $\mathsf{FL}(\pi)$ by applying it to $w^{\ast}(\pi,\overline{w})$, using the alphabet $\overline{\mathbb{Z}}_{+}$. The fixed point of $\varphi$ is the unique fall-labeling of $\pi$ such that $w^{\ast}(\pi,\overline{w})=\overline{1}\,\overline{2}\cdots\overline{k}$.

Proposition 4.14**.**

For $\pi\in\mathsf{LRP}(m+k,n+k)_{\ast k}$, and for $\overline{w}\in\mathsf{FL}(\pi)$, we have

[TABLE]

Proof.

By definition, this is equivalent to saying that $\mathsf{tdinv}(\psi^{-1}(\pi,\overline{w}))=\mathsf{tdinv}(\psi^{-1}(\pi,\varphi(\overline{w})))$. By the same argument as [GillespieGorskyGriffin2025]*Lemma 4.6, the result follows. ∎

5. Computing the dinv

This section is dedicated to the proof of the following theorem.

Theorem 5.1**.**

Let $\pi\in\mathsf{LRP}(m+k,n+k)_{\ast k}$, and let $\overline{w}\in\mathsf{FL}(\pi)$ be the fall-labeling such that $w^{\ast}(\pi,\overline{w})=\overline{1}\,\overline{2}\cdots\overline{k}$. Let $\tilde{\pi}=\psi^{-1}(\pi,\overline{w})$. Then

[TABLE]

Proof.

First, notice that if $k=0$, then $\psi$ is the identity and there is nothing to prove, so from now on, assume $k>0$.

We use plain letters to denote sets referring to the path $\pi$ (and to its fall-labeling $\overline{w}$); sets referring to $\psi^{-1}(\pi,\overline{w})$ are denoted with a tilde. For example, we set $C_{+}\coloneqq C_{+}(\pi)$, $\tilde{C}_{-}\coloneqq C_{-}(\psi^{-1}(\pi,\overline{w}))$, and so on. Using the ENS representation for both paths, we can think of $\tilde{\pi}$ as the path consisting of the steps $\mathcal{H}\sqcup\mathcal{V}\sqcup\mathcal{D}\sqcup\mathcal{B}\sqcup\mathcal{S}$ and of $\pi$ as the subpath consisting of the steps $\mathcal{H}\sqcup\mathcal{V}\sqcup\mathcal{D}$ (see Figure 9).

By definition, we have $\mathsf{dinv}(\pi)=\mathsf{tdinv}(\pi)+\mathsf{cdinv}(\pi)$ and $\mathsf{dinv}(\tilde{\pi})=\mathsf{tdinv}(\tilde{\pi})+\mathsf{cdinv}(\tilde{\pi})$. By Remark 4.11, we have that $\mathsf{tdinv}(\tilde{\pi})=\mathsf{tdinv}(\pi)+\mathsf{falldinv}(\pi,\overline{w})$, so it remains to prove the following:

[TABLE]

Since $k>0$, we have $m<n+km$, and because the decorated steps do not contribute to $\mathsf{cdinv}(\tilde{\pi})$, the only non-zero summands in $\mathsf{cdinv}(\tilde{\pi})$ using Definition 3.18 are $\#\tilde{C}_{-}$ and $\#\tilde{B}$. Partition $\tilde{C}_{-}$ and $\tilde{B}$ based on the contributions of $\mathcal{V}$ (steps with small labels) and $\mathcal{B}$ (steps with big labels):

[TABLE]

Note that the definitions of $\tilde{C}_{\text{s}}$ and $\tilde{C}_{\text{b}}$ have a $+k$ term due to the fact that the diagonal of $\tilde{\pi}$ has slope $\frac{n}{m}+k$ and not $\frac{n}{m}$; in the ENS representation, this means that we are counting the pairs $(i,j)\in\mathcal{V}\times\mathcal{H}$ (resp. $\mathcal{B}\times\mathcal{H}$) such that both endpoints of step $i$ diagonally project onto step $j$ or onto any of the $k$ South steps immediately preceding it (right-most endpoint included, left-most endpoint excluded).

Partition also the contributions to $\mathsf{falldinv}(\pi,\overline{w})$ as follows:

[TABLE]

The set $F_{\text{s}}^{\swarrow}$ (resp. $F_{\text{b}}^{\swarrow}$) consists of all pairs of vertical steps $(i,j)$ such that step $i$ has a small (resp. big) label, step $j$ has a big label, and $i$ attacks $j$ from the right. The set $F^{\nearrow}$ consists of all pairs of vertical steps $(i,j)$ where $j$ has a big label and $i$ attacks $j$ from the left.

We can rewrite the statement as

[TABLE]

Note that $\tilde{B}_{\text{s}}=B$, so the statement simplifies to

[TABLE]

In $F_{\text{s}}^{\swarrow}$, group the steps with a big label by the horizontal step following them, obtaining the set

[TABLE]

with $\#F_{\text{s}}^{\prime}=\#F_{\text{s}}^{\swarrow}$. Now, the two unions $C_{+}\cup\tilde{C}_{\text{s}}$ and $C_{-}\cup F_{\text{s}}^{\prime}$ are disjoint unions, and they are both equal to

[TABLE]

Therefore, we can cancel out the corresponding four summands from Equation 3 and reduce the statement to

[TABLE]

For $s\in[\overline{k}]$, let $\pi_{s}$ be the subpath obtained from the ENS representation of $\tilde{\pi}$ by removing all steps $i\in\mathcal{B}$ with $w_{i}\geq s$ and the corresponding steps $i^{\prime}\in\mathcal{S}$ (recall that there is a natural bijection between $\mathcal{B}$ and $\mathcal{S}$ sending $i\in\mathcal{B}$ to the step $i^{\prime}\in\mathcal{S}$ with the same endpoints). Note that $\pi_{\overline{1}}=\pi$. These paths are depicted in Figure 10.

For $i\in\mathcal{B}$, denote by $\ell_{i}$ the half-line parallel to the main diagonal (in the ENS representation) ending at the bottom-most endpoint of step $i$. We are going to show that the left-hand side of Equation 4 counts the number of times $\ell_{i}$ crosses the path $\pi_{w_{i}}$ (summed over all $i\in\mathcal{B}$), where crossing a step in $\mathcal{V}\sqcup\mathcal{B}$ counts as $-1$ and crossing every other step counts as $+1$. For the purpose of intersecting $\ell_{i}$, vertical steps have their top-most endpoint excluded and their bottom-most endpoint included; horizontal steps have their left-most endpoint excluded and their rightmost endpoint included. Since $\ell_{i}$ ends weakly above the path, the total count for $\ell_{i}$ must be [math] if $\ell_{i}$ starts weakly above the path and $+1$ if it starts strictly below the path; the result (summed over all $i\in\mathcal{B}$) is precisely counted by $\tilde{B}_{\text{b}}$, which appears on the right-hand side of Equation 4.

The attacking pairs $j\rightarrow i$ in $F^{\nearrow}$ correspond exactly to the intersections of the half-lines $\ell_{i}$ with the vertical steps $j\in\mathcal{V}\sqcup\mathcal{B}$ belonging to $\pi_{w_{i}}$. Therefore, it remains to show that $\#C^{*}+\#\tilde{C}_{\text{b}}-\#F_{\text{b}}^{\swarrow}$ counts the intersections of the half-lines $\ell_{i}$ with the steps in $\mathcal{H}\sqcup\mathcal{D}\sqcup\mathcal{S}$ belonging to $\pi_{w_{i}}$, or equivalently, with the steps in $\mathcal{H}\sqcup\mathcal{D}\sqcup\{i^{\prime}\in\mathcal{B}\mid w_{i^{\prime}}<w_{i}\}$.

Fix $i\in\mathcal{V}$ and $j\in\mathcal{H}$ with $j<i$. Denote by $r_{ij}^{\prime}$ the number of decorated steps $i^{\prime}\in\mathcal{D}$ immediately preceding step $j$ with $\overline{w}(i^{\prime})\leq w_{i}$. If the step with fall-labeling equal to $w_{i}$ is among these, denote it by $i_{*}$ (recall that there is exactly one such step in the entire path $\pi$). Note that $v_{i_{*}}=v_{j}+\frac{n}{m}+r_{ij}^{\prime}-1$ by definition of $i_{*}$. Let $r_{ij}^{\prime\prime}=1$ if $i_{*}$ is defined and [math] otherwise.

We need to count whether the half-line $\ell_{i}$ intersects step $j$, the $r_{j}$ decorated steps immediately before it, or the $w_{i}-r_{j}+r_{ij}^{\prime}-1$ vertical steps with big labels $<w_{i}$ immediately above. This happens if and only if

[TABLE]

Here we treat $w_{i}$ as a positive integer, even though technically $w_{i}\in\overline{\mathbb{Z}}_{+}$. In $-\#F_{\text{b}}^{\swarrow}$, we get a contribution of $-1$ if step $i$ attacks any of the steps in $\mathcal{B}$ above the left-most endpoint of step $j$, which happens if and only if

[TABLE]

If $r_{ij}^{\prime\prime}=1$ (so $i_{*}$ is defined), look at contributions to $C^{*}$ from pairs $(i_{*},j^{\prime})$ where $j^{\prime}>i$ has at least one endpoint below $i$. If $(i_{*},j^{\prime})$, the decorated steps immediately before $j^{\prime}$ have a label $<w_{i}$, whereas all decorated steps immediately after $j^{\prime}$ (including $j^{\prime}$) have a label $>w_{i}$, by construction of $\overline{w}$. Therefore, step $j^{\prime}$ is uniquely determined by $v_{i}=v_{j^{\prime}}^{+}+w_{i}-1$, and by definition of $C^{*}$, we have $(i_{*},j^{\prime})\in C^{*}$ if and only if $v_{i_{*}}\leq v_{j^{\prime}}^{+}<v_{i_{*}}+1$; this condition can be rewritten as

[TABLE]

To conclude, the contributions of $\#C^{*}+\#\tilde{C}_{\text{b}}-\#F_{\text{b}}^{\swarrow}$ relevant to the pair $(i,j)$ cancel out except if Equation 5 holds, in which case there is a contribution of +1, as desired. ∎

6. Proof of Theorem 1.1

It is now time to put together the pieces and conclude the proof of Theorem 1.1. First, we recall the statement of the rational shuffle theorem.

Theorem 6.1** (Rational shuffle theorem [Mellit2021Rational]).**

For $m,n\in\mathbb{N}$, we have

[TABLE]

Applying the skewing operator $s_{(m-1)^{k}}^{\perp}$ to the combinatorial interpretation of $e_{m,n+km}$ given by Theorem 6.1, and using Proposition 4.4, we obtain

[TABLE]

Combining this with Proposition 4.9, we get

[TABLE]

By Proposition 4.14 and Theorem 5.1, this equality becomes

[TABLE]

which is exactly Theorem 1.1, as desired.

Let us recall the statement of the rectangular shuffle conjecture.

Conjecture 6.2** ([IraciPagariaPaoliniVandenWyngaerd2023]*Conjecture 4.2).**

For $m,n\in\mathbb{N}$ and $d=\gcd(m,n)$, we have

[TABLE]

Using the fact that $\gcd(m,n+km)=\gcd(m,n)$, the previous argument also proves Theorem 1.2. In particular, due to [IraciPagariaPaoliniVandenWyngaerd2023]*Theorem 6.1, the equality of Theorem 1.2 holds when $d=1$.

7. Links with the existing literature

The objects and the symmetric functions introduced in this paper reduce to several other families of objects and symmetric functions in the literature, for suitable choices of $m$, $n$, and $k$. In this section, we give a non-exhaustive overview of results that can be deduced from the previous sections, together with several new interesting conjectures.

7.1. $D_{\alpha}$ operators and $e$-positivity

Let $\alpha\vDash n$ defined by $\alpha_{i}\coloneqq\left\lceil\frac{ni}{m}\right\rceil-\left\lceil\frac{n(i-1)}{m}\right\rceil$, and let $x^{\alpha}\coloneqq\prod_{i}x_{i}^{\alpha_{i}}$. Recall that, in the notation of [BlasiakHaimanMorsePunSeelinger2023ShuffleAnyLine] (cf. [BlasiakHaimanMorsePunSeelinger2023ProofExtendedDelta, GillespieGorskyGriffin2025]), we have the identity

[TABLE]

Then, by [GillespieGorskyGriffin2025]*Lemma 3.12, we can deduce the following identity.

Proposition 7.1**.**

For $m,n,k\in\mathbb{N}$ with $m>0$, and $\alpha$ as above, we have

[TABLE]

Since the operator $H_{q,t}^{m}\left(\cdot\right)_{\textnormal{pol}}$ is linear, we can split the sum into single summands. The monomials in $h_{k}(x_{1}\cdots x_{m})$ are naturally indexed by weak compositions $\beta\vDash k$ of length $\ell(\beta)\leq m$, where $\beta_{i}$ is the exponent of $x_{i}$.

Definition 7.2**.**

Let $(\pi,\mathcal{D},w)\in\mathsf{LRP}(m+k,n+k)_{\ast k}$. We define its fall-composition $\beta(\pi)\vDash k$ as $\beta(\pi)\coloneqq(r_{i})_{i\in\mathcal{H}}$, that is, $\beta_{i}$ is the number of decorated falls immediately preceding the $i\textsuperscript{th}$ non-decorated horizontal step of $\pi$.

We can naturally split the generating function of $\mathsf{LRD}(m+k,n+k)_{\ast k}$ according to $\beta(\pi)$. Ideally, we would like to match each summand to a monomial in $h_{k}(x_{1},\dots,x_{m})$. This is not possible in general because for a generic $\beta$ the symmetric function $D_{\alpha+\beta}(1)$ is not monomial-positive. However, if we set $q=1$, we get the following.

Theorem 7.3**.**

For $m,n,k\in\mathbb{N}$ with $m>0$, $\alpha$ as above, and $\beta\vDash k$ of length $\ell(\beta)\leq m$, we have

[TABLE]

Theorem 7.3 is essentially a reformulation of [BlasiakHaimanMorsePunSeelinger2023ShuffleAnyLine]*Equation (148), which states

[TABLE]

where $\lambda$ is any path above the one prescribed by $\gamma$ (that is, the one with $\gamma_{i}$ vertical steps on the vertical line $x=i-1$), $a(\lambda)$ is the area enclosed between the two paths, and $b(\lambda)$ is the composition such that $b(\lambda)_{i}$ is the number of vertical steps of $\lambda$ on the vertical line $x=i-1$. In order to derive Theorem 7.3, it is sufficient to contract the columns containing decorated falls. The area stays the same, and the obtained path is one of the paths above the shape prescribed by $\gamma$. This correspondence is trivially bijective once one prescribes the positions of the decorated falls.

However, this is interesting because this formula refines the fall Delta theorem and our rational extension in a way that was not, to the authors’ awareness, known before. It would be interesting to find a formula for such summands that refines $\Delta^{\prime}_{e_{n}}e_{n+k}=\Theta_{e_{k}}\nabla e_{n}$ in some way. Indeed, Theorem 7.3 has the same combinatorial flavor as [HicksRomero2018Polyominoes]*Theorem 1.1 (cf. [IraciRomero2022DeltaTheta]*Theorem 8.6), as they both split a positive generating function into non-positive pieces that become positive and combinatorial when specializing one variable to $1$.

7.2. Theta operators

Another attempt at finding a decorated extension of the rational shuffle theorem is [IraciPagariaPaoliniVandenWyngaerd2023]*Conjecture 4.4, which states the following.

Conjecture 7.4**.**

For any $m,n\in\mathbb{N}$, we have

[TABLE]

where $\mathsf{LRD}(m+k,n+k)^{\ast k}$ denotes the set of rise-decorated labeled rectangular Dyck paths of size $(m+k)\times(n+k)$.

The main difference with Theorem 1.1 consists in having decorations on the rises rather than the falls. The problem of finding a $\mathsf{dinv}$ statistic for rise-decorated paths remains open. While the distinction is almost irrelevant when $m=n$, when $m\neq n$ we get really different sets of objects. However, the distinction disappears again when looking at non-labeled paths, or even Schröder paths, as in both cases the reflection with respect to the antidiagonal yields a bijection between the sets of objects.

It is known that, in order to extract Schröder paths, one has to take the scalar product with $h_{d}e_{n+k-d}$, where $d$ is the number of diagonal steps. In terms of labelings, taking such a scalar product corresponds to picking $d$ peaks of the path labeled with a maximal label, and then labeling the rest of the path in increasing order with respect to the distance from the diagonal (taken vertically or horizontally depending on the set of objects involved). In particular, this suggests the following symmetric function identity, backed up by computer experiments.

Conjecture 7.5**.**

For $m,n,k,d\in\mathbb{N}$ with $m>0$ and $d\leq\min\{m+k,n+k\}$, we have

[TABLE]

In principle, this identity could be exploited to find a $\mathsf{dinv}$ statistic on rise-decorated rectangular Dyck paths. While our first attempts have not been successful, this warrants further investigation.

7.3. The Delta square conjecture

In [DAdderioIraciVandenWyngaerd2019DeltaSquareConjecture], the authors conjecture a decorated version of the square paths theorem, in term of rise-decorated square paths. We refer to [DAdderioIraciVandenWyngaerd2019DeltaSquareConjecture, IraciVandenWyngaerd2021ValleySquare] for all missing definitions. The statement is the following.

Conjecture 7.6** ([DAdderioIraciVandenWyngaerd2019DeltaSquareConjecture]*Conjecture 3.12).**

For $n,k\in\mathbb{N}$, we have

[TABLE]

where $\mathsf{LRP}(n+k,n+k)^{\ast k}$ denotes the set of rise-decorated labeled rectangular Dyck paths of size $(n+k)\times(n+k)$.

Notice the extra factor $[n]_{t}/[n+k]_{t}$ in the expression. In [IraciVandenWyngaerd2021ValleySquare], the authors partly address this issue. Via the identity

[TABLE]

and the $q,t$-reversed version

[TABLE]

the authors formulate a variant of the conjecture in terms of valley-decorated square paths.

A contractible valley of a labeled rectangular path is a vertical step that is preceded by either two horizontal steps, or one horizontal step such that the label of the vertical step immediately before such step is strictly smaller than the label of the vertical step immediately after it. These kinds of decorations first appeared in the valley version of the Delta conjecture [HaglundRemmelWilson2018DeltaConjecture]. The authors give the two following conjectures.

Conjecture 7.7** ([IraciVandenWyngaerd2021ValleySquare]*Conjecture 5).**

For $n,k\in\mathbb{N}$, we have

[TABLE]

where $\mathsf{LRP}(n+k,n+k)^{\bullet k}$ denotes the set of valley-decorated labeled rectangular Dyck paths of size $(n+k)\times(n+k)$.

Conjecture 7.8** ([IraciVandenWyngaerd2021ValleySquare]*Conjecture 8).**

For $n,k\in\mathbb{N}$, we have

[TABLE]

where $\mathsf{LRP}^{\prime}(n+k,n+k)^{\bullet k}$ is the subset of $\mathsf{LRP}(n+k,n+k)^{\bullet k}$ where the bottom-most vertical step of the path among those lying on the base diagonal is not decorated.

Notice that 7.8 does not have any multiplicative factor. Indeed, even when $m=n$, the combinatorial symmetry of rises and falls does not hold for generic rectangular paths, because of the condition of ending with a horizontal step: since they might start with a horizontal step, the symmetry with respect to the antidiagonal does not preserve the set.

This suggests that decorating rises might not be the correct approach, and we should look at falls instead. Indeed, computer experiments suggest the following.

Conjecture 7.9** (Delta square conjecture, fall version).**

For $n,k\in\mathbb{N}$ with $n>0$, we have

[TABLE]

Since we already have a formula for the right-hand side of 7.9, we also conjecture the following.

Conjecture 7.10**.**

For $n,k\in\mathbb{N}$, we have

[TABLE]

Acknowledgements

We would like to thank Matteo Migliorini for an insight that led us to the ENS representation for our objects. We acknowledge the support of INDAM-GNSAGA grant CUP-E53C23001670001 and MUR grant PRIN 2022A7L229 “Algebraic and topological combinatorics” CUP-J53D23003660006.