3.1 Rectangular paths

The restriction of ending with a horizontal step is given in analogy with the original definition of a labeled square path [Reference Loehr and WarringtonLW07] and also because the resulting combinatorics better matches the algebraic expressions. We denote the sets of rectangular paths of size $m \times n$ as $\operatorname {RP}(m,n)$ .

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).

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

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

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

Figure 2

A $7 \times 9$ rectangular path with its base diagonal (dashed) and main diagonal (solid).

3.2 Decorated rectangular paths

In a similar fashion to the rise version of the Delta conjecture [Reference Haglund, Remmel and WilsonHRW18] (which is now a theorem [Reference D’Adderio and MellitDM22, Reference Blasiak, Haiman, Morse, Pun and SeelingerBHM+23a]), we introduce the concept of decorated falls for rectangular paths.

This definition is analogous to [Reference Iraci, Pagaria, Paolini and Vanden WyngaerdIPPVW23, 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 $0 \leq x \leq m + k$ diagonally projects onto a horizontal or vertical step if the vertical translation of the broken diagonal passing through $(x, y)$ intersects the step. Formally, if the broken diagonal is denoted by $\mathrm {BD}$ , there exists a unique $v \in \mathbb {R}$ such that $(x,y) \in \mathrm {BD}+ (0,v)$ ; the point $(x,y)$ diagonally projects onto a step s if $s \cap (\mathrm {BD}+ (0,v)) \neq \varnothing $ .

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

Figure 3

On the left, a decorated rectangular Dyck path of size $(6 + 3) \times (3+3)$ with its broken diagonal. On the right, the ENS representation of the same path. Decorated steps are highlighted in dark red. The three squares contributing to the area are highlighted in gray.

Figure 3 Long description

The left panel shows a rectangular grid with a blue Dyck path from the bottom-left to the top-right, staying above a broken diagonal. Three horizontal steps are marked in dark red with asterisks above them. Three gray squares beneath the path and above the diagonal indicate the area. The right panel presents the ENS representation on a smaller grid, where the same three decorated steps are now vertical, marked in dark red with asterisks beside them. The corresponding area squares are shaded gray below the path. Both panels use a grid structure, and the decorated steps are consistently highlighted.

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. In addition, 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.

Figure 4

A $9 \times 7$ labeled rectangular path (left) and labeled decorated Dyck path (right).

Figure 4 Long description

The left panel shows a 9 by 7 grid with a blue path starting at the bottom-left corner, moving right and up in steps. The vertical steps are labeled from bottom to top as 2, 7, 4, and the horizontal steps as 1, 4, 2, 7, 4. The path follows the grid lines and stays above a diagonal gray line from bottom-left to top-right. The right panel shows a similar grid and path, but the vertical steps are labeled 4, 2, 4, and the horizontal steps are labeled 1, 7, 1. Three asterisks are placed above the horizontal segment labeled 7, and one asterisk is above the final horizontal segment labeled 1. The path also stays above a diagonal gray line.

We say that a labeling is standard if the set of labels is $[n+k] {:=}q \{1, \dots , n+k\}$ , 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 $\operatorname {LRP}(m,n)$ and $\operatorname {LRD}(m,n)$ 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 $\operatorname {LRP}(m+k,n+k)_{\ast k}$ and $\operatorname {LRD}(m+k,n+k)_{\ast k}$ , respectively.

It will be useful later to consider labelings in the alphabet

$$\begin{align*}\mathbb{Z}_+ \cup \overline{\mathbb{Z}}_+ = 1 < 2 < 3 < \dots < \overline{1} < \overline{2} < \overline{3} < \dots. \end{align*}$$

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 [Reference Iraci, Pagaria, Paolini and Vanden WyngaerdIPPVW23] (cf. [Reference Bergeron, Garsia, Sergel Leven and XinBGSLX16, Reference Hicks and SergelHS15, Reference MellitMel21]) 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

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

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

$$\begin{align*}\operatorname{tdinv}(\pi, w) {:=}q \# \{ i, j \in \mathcal{V} \mid w_i < w_j \text{ and } i \to j \}. \end{align*}$$

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 [Reference Hicks and SergelHS15] for Dyck paths and later in [Reference Iraci, Pagaria, Paolini and Vanden WyngaerdIPPVW23] for rectangular paths.

First, we introduce some notation.

We are now ready to define the dinv correction (or cdinv). We give two different definitions, which we prove to be equivalent in Proposition 3.20.

Definition 3.17. Let $(\pi , \mathcal {D})$ be a fall-decorated rectangular path, and let

$$ \begin{align*} D_+ & {:=}q \left\{ (i, j) \in \mathcal{V} \times \mathcal{H} \mid j < i \text{ and } v_i < v_j \leq v_i + 1 - \textstyle\frac{n}{m} \right\} \\ D_- & {:=}q \left\{ (i, j) \in \mathcal{V} \times \mathcal{H} \mid j < i \text{ and } v_j \leq v_i < v_j + \textstyle\frac{n}{m} - 1 \right\} \\ D_+^\ast & {:=}q \left\{ (i, j) \in \mathcal{D}\times \mathcal{H} \mid i < j \text{ and } v_i \leq v_j < v_i + 1- \textstyle\frac{n}{m} \right\} \\ D_-^\ast & {:=}q \left\{ (i, j) \in \mathcal{D} \times \mathcal{H} \mid i < j \text{ and } v_j < v_i \leq v_j - 1+ \textstyle\frac{n}{m} \right\} \\ B & {:=}q \{ i \in \mathcal{V} \mid v_i < 0 \} \\ B^* &{:=}q \{ i \in \mathcal{D} \mid v_i \leq 0 \}. \end{align*} $$

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

$$\begin{align*}\operatorname{cdinv}(\pi, \mathcal{D}) {:=}q \# D_+ - \# D_- + \# D_+^\ast -\# D_-^\ast +\# B -\# B^\ast. \end{align*}$$

The set $D_+$ in Definition 3.17 consists of all pairs of vertical and horizontal steps $(i, j)$ such that the vertical step appears 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 $D_{-}^{\ast }$ .

Figure 5

Graphical description of the pairs of steps in $D_+$ (top left), $D_-$ (top right), $D_+^\ast $ (bottom left), and $D_-^\ast $ (bottom right). The dashed lines denote diagonal projections: in particular, they are vertical translates of the broken diagonal and might not be straight.

Figure 5 Long description

Top left panel shows two horizontal segments connected by two dashed diagonal lines to a vertical segment ending in an open circle. Top right panel has a horizontal segment with endpoints marked by open and filled circles, connected by dashed lines to a vertical segment. Bottom left panel displays a horizontal segment with an asterisk at the left endpoint and an open circle at the right, connected by dashed lines to a vertical segment. Bottom right panel shows a horizontal segment with an asterisk at the left endpoint and a filled circle at the right, connected by dashed lines to a vertical segment. All dashed lines represent vertical translations of a broken diagonal.

Figure 6

Graphical description of the pairs of steps in $D_+^\ast $ (left), and $D_-^\ast $ (right) in the ENS representation.

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

$$ \begin{align*} C_+ & {:=}q \left\{ (i, j) \in \mathcal{V} \times \mathcal{H} \mid j < i, \; r_j = 0, \text{ and } v_i < v_j \leq v_i + 1 - \textstyle\frac{n}{m} \right\} \\ C_- & {:=}q \left\{ (i, j) \in \mathcal{V} \times \mathcal{H} \mid j < i \text{ and } v_j \leq v_i < v_j + \textstyle\frac{n}{m} + r_j - 1 \right\} \\ C^\ast & {:=}q \left\{ (i, j) \in \mathcal{D} \times (\mathcal{H} \sqcup \mathcal{D}) \mid i < j-1 \text{ and } v_i \leq v_j^+ < v_i + 1 \right\} \end{align*} $$

where $v_j^+ {:=}q v_j + \chi _{\mathcal {D}}(j) + \textstyle \frac {n}{m} \chi _{\mathcal {H}}(j)$ 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. Here, $\chi _{A}$ denotes the indicator function of the set A.

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

$$\begin{align*}\operatorname{cdinv}(\pi, \mathcal{D}) {:=}q \# C_+ - \# C_- + \# C^\ast + \# B. \end{align*}$$

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 nondecorated.

The set $C^\ast $ consists of all pairs of nonconsecutive 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 a path $(\pi , \mathcal {D}) \in \operatorname {RP}(m+k,n+k)_{*k}$ , let

$$\begin{align*}E = \left\{ (i, j) \in \mathcal{V} \times \mathcal{D} \mid j < i \text{ and } v_i < v_j \leq v_i + 1 \right\}, \end{align*}$$

that is, the set of pairs of vertical and horizontal steps $(i, j)$ such that the horizontal step is decorated, the vertical step appears after the horizontal step, and the right-most endpoint of the horizontal step diagonally projects onto the vertical step (top-most endpoint excluded, bottom-most endpoint included). Then

$$\begin{align*}\# E = \sum_{\substack{(i,j) \in \mathcal{V} \times \mathcal{H} \\ j < i}} \chi_{[\frac{n}{m}-1,\frac{n}{m}+r_j-1)}(v_i - v_j). \end{align*}$$

Figure 7

Graphical description of the pairs of steps in $C_+$ (top), $C_-$ (bottom left), and $C^\ast $ (bottom right). The dashed lines denote the diagonal projections: in particular they are vertical translates of the broken diagonal and might not be straight.

Figure 7 Long description

The top panel represents C plus. On the left, a horizontal segment is labeled with dotted lines and the text ‘no asterisk’ above it. Dashed lines project diagonally upward and rightward to a vertical segment ending in a filled circle at the top and an open circle at the bottom. The bottom left panel shows C minus. A horizontal segment at the bottom is marked with three asterisks and ends in an open circle on the left and a filled circle on the right. Dashed lines project diagonally upward and rightward to a vertical segment. The bottom right panel shows C star. A horizontal segment at the bottom left is marked with an asterisk and ends in an open circle on the left and a filled circle on the right. Dashed lines project diagonally upward and rightward to a horizontal segment at the top right, also marked with an asterisk. All dashed lines represent vertical translations of a broken diagonal, not necessarily straight.

Figure 8

Graphical description of the pairs of steps in $C_-$ (left) and $C^\ast $ (center and right) in the ENS representation.

Figure 8 Long description

From left to right, the first panel shows a vertical solid line with an open circle at the top, three asterisks along the line, a right-angle corner at the bottom, and a horizontal solid line ending in a filled circle. A dashed line connects the open circle to the filled circle. The center panel shows a vertical solid line with an open circle at the top, one asterisk, a right-angle corner, and a horizontal solid line ending in a filled circle. A dashed line connects the open circle to the filled circle. The right panel shows a vertical solid line with an open circle at the top, one asterisk, and a filled circle at the bottom. A dashed line connects the open circle to the filled circle. Each panel represents a different step configuration in the E N S representation.

Proof. Let $(i,j) \in \mathcal {V} \times \mathcal {H}$ , suppose that $r_j \neq 0$ , and take $1 \leq p \leq r_j$ . By construction, we have $j-p \in \mathcal {D}$ , and $j < i$ holds if and only if $j-p < i$ , in which case $v_{j-p} = v_j + \frac {n}{m} + p - 1$ . Therefore

$$\begin{align*}\chi_{[-1,0)}(v_i-v_{j-p}) = \chi_{[\frac{n}{m}+p-2,\frac{n}{m}+p-1)}(v_i - v_j). \end{align*}$$

Now,

$$ \begin{align*} \# E & = \sum_{\substack{(i,j) \in \mathcal{V} \times \mathcal{D} \\ j < i}} \chi_{[-1,0)}(v_i-v_j) \\ & = \sum_{\substack{(i,j) \in \mathcal{V} \times \mathcal{H} \\ j < i}} \, \sum_{p=1}^{r_j} \chi_{[\frac{n}{m}+p-2,\frac{n}{m}+p-1)}(v_i - v_j) \\ & = \sum_{\substack{(i,j) \in \mathcal{V} \times \mathcal{H} \\ j < i}} \chi_{[\frac{n}{m}-1,\frac{n}{m}+r_j-1)}(v_i - v_j).\\[-42pt] \end{align*} $$

Proposition 3.20. For each path $(\pi , \mathcal {D}) \in \operatorname {RP}(m+k,n+k)_{*k}$ we have

$$\begin{align*}\# D_+ - \# D_- + \# D_+^\ast -\# D_-^\ast -\# B^\ast = \# C_+ - \# C_- + \# C^\ast.\end{align*}$$

In particular, Definitions 3.17 and 3.18 are equivalent.

Proof. Define E as in Lemma 3.19. The statement is implied by the following two equalities:

$$\begin{align*}\# D_- + \#E = \# D_+ - \# C_+ + \# C_- \quad \text{and} \quad \# C^\ast - \#D_+^\ast + \#D_-^\ast = \# E - \# B^\ast. \end{align*}$$

Consider the first equality. By Definitions 3.17 and 3.18 and Lemma 3.19, we have

(3.1) $$ \begin{align} \# D_- & = \sum_{\substack{(i,j) \in \mathcal{V} \times \mathcal{H} \\ j < i}} \chi_{[0,\frac{n}{m}-1)}(v_i-v_j) \end{align} $$

(3.2) $$ \begin{align} \# E & = \sum_{\substack{(i,j) \in \mathcal{V} \times \mathcal{H} \\ j < i}} \chi_{[\frac{n}{m}-1,\frac{n}{m}+r_j-1)}(v_i - v_j) \end{align} $$

(3.3) $$ \begin{align} \# D_+ - \# C_+ &= \sum_{\substack{(i,j) \in \mathcal{V} \times \mathcal{H} \\ j < i}} (1 - \delta_{r_j,0}) \, \chi_{[\frac{n}{m}-1,0)}(v_i-v_j) \end{align} $$

(3.4) $$ \begin{align} \# C_- & = \sum_{\substack{(i,j) \in \mathcal{V} \times \mathcal{H} \\ j < i}} \chi_{[0,\frac{n}{m}+r_j-1)}(v_i-v_j). \end{align} $$

Now, fix a pair $(i,j) \in \mathcal {V} \times \mathcal {H}$ with $j < i$ . If $r_j = 0$ , then the corresponding terms in (3.2) and (3.3) vanish, and the terms in (3.1) and (3.4) are equal. If $r_j \neq 0$ and $n < m$ , the term in (3.1) vanishes, and the term in (3.2) is the sum of the terms in (3.3) and (3.4). If $r_j \neq 0$ and $n \geq m$ , the term in (3.3) vanishes, and the term in (3.4) is the sum of the terms in (3.1) and (3.2). This proves the first equality.

Consider now the second equality. Let

$$\begin{align*}F = \left\{ (i, j) \in \mathcal{D} \times (\mathcal{H} \sqcup \mathcal{D}) \mid i < j-1 \text{ and } v_j < v_i \leq v_j^+ \right\}, \end{align*}$$

and note that

$$\begin{align*}F \setminus D^\ast_- = \left\{ (i, j) \in \mathcal{D} \times (\mathcal{H} \sqcup \mathcal{D}) \mid i < j-1 \text{ and } v_j < v_i \leq v_j^+ < v_i + 1 \right\} = C^\ast \setminus D_+^\ast. \end{align*}$$

Since $D_-^\ast \subseteq F$ and $D_+^\ast \subseteq C^\ast $ , we have $\# F = \# C^\ast - \#D_+^\ast + \#D_-^\ast $ and so it is enough to show that $\# E = \# F + \# B^\ast $ . Let $(i,j) \in E$ ; this means that the vertical translate of the broken diagonal passing through the right-most endpoint of j crosses the vertical step i (top-most endpoint excluded, bottom-most endpoint included) at height $v_j$ .

If the same vertical translate of the broken diagonal crosses another horizontal step, let $j'> i > j$ be the smallest such step; then $(j,j') \in F$ and this correspondence is one-to-one. If it does not cross any other horizontal step, then we necessarily have $v_j \leq 0$ , and for each j such that $v_j \leq 0$ there exists exactly one i with this property, namely $\max \{ i \in \mathcal {V} \mid (i,j) \in E \}$ . The second equality follows.

Definition 3.21. We define the dinv of a labeled rectangular path $(\pi , w)$ as

$$\begin{align*}\operatorname{dinv}(\pi, w) {:=}q \operatorname{tdinv}(\pi, w) + \operatorname{cdinv}(\pi). \end{align*}$$

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

4.1 Combinatorial interpretation of the skewing operator

Recall that

(4.1) $$ \begin{align} g[Y]^\perp f[X+Y] \big|_{Y=0} = g[X]^\perp f[X] \end{align} $$

for any symmetric functions $f, g \in \Lambda $ . Readers unfamiliar with plethystic notation can refer to [Reference Loehr and RemmelLR11] for details (cf. [Reference MacdonaldMac95]); using the results in [Reference Loehr and RemmelLR11, Section 3.2], (4.1) is immediate when $f, g$ are Schur functions, and holds by linearity in the general case.

Let $\alpha $ be any weak composition. We want to apply the skewing operator $h_\alpha ^\perp $ to the expression

$$\begin{align*}\sum_{\pi \in \operatorname{LRP}(m,n)} q^{\operatorname{dinv}(\pi)} t^{\operatorname{area}(\pi)} x^\pi. \end{align*}$$

This operation is well-defined as the expression is a positive sum of LLT polynomials, and so it is a (Schur positive) symmetric function. Using (4.1), we know that

$$\begin{align*}h_\alpha^\perp \sum_{\pi \in \operatorname{LRP}(m,n)} q^{\operatorname{dinv}(\pi)} t^{\operatorname{area}(\pi)} x^\pi = h_\alpha[Y]^\perp \left. \left( \sum_{\pi \in \operatorname{LRP}(m,n)} q^{\operatorname{dinv}(\pi)} t^{\operatorname{area}(\pi)} x^\pi \right)[X+Y] \right\vert_{Y=0}. \end{align*}$$

The plethystic substitution in the combinatorial term corresponds to the replacement of the alphabet $\mathbb {Z}_+$ with $\mathbb {Z}_+ \cup \overline {\mathbb {Z}}_+$ in the labeling, where we can assume that the labels in $\mathbb {Z}_+$ are smaller than the labels in $\overline {\mathbb {Z}}_+$ , since the expression is symmetric.

Now, since homogeneous and monomial symmetric functions are dual to each other, the skewing operator $h_\alpha [Y]^\perp $ followed by the substitution $Y=0$ isolates the terms of the form $f[X] m_\alpha [Y]$ for some $f \in \Lambda $ ; since the formula is symmetric in Y, the coefficient of $m_\alpha [Y]$ is the same as that of $y^\alpha $ when the expression is interpreted as a formal power series in Y with coefficients in $\Lambda $ .

For analogous reasons, the same holds for $\operatorname {LRD}(m,n)$ . This motivates the following definition.

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

$$\begin{align*}h_\alpha^\perp \sum_{\pi \in \operatorname{LRP}(m,n)} q^{\operatorname{dinv}(\pi)} t^{\operatorname{area}(\pi)} x^\pi = \sum_{\pi \in \operatorname{LRP}(m,n)^{\alpha}} q^{\operatorname{dinv}(\pi)} t^{\operatorname{area}(\pi)} x^\pi, \end{align*}$$

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

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

Definition 4.3 ([Reference Gillespie, Gorsky and GriffinGGG25, 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

$$\begin{align*}\mathrm{sgn}(\alpha) {:=}q \sum_{\alpha_i> 0} (\alpha_i - 1) = \# \{1 \leq i \leq k \mid \alpha_i = 0\}. \end{align*}$$

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

Proposition 4.5 [Reference Gillespie, Gorsky and GriffinGGG25, Equation (10)]

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

$$\begin{align*}s_{(m-1)^k} = \sum_{\alpha \text{ allowable}} (-1)^{\mathrm{sgn}(\alpha)} h_{\tilde\alpha}\quad \pmod{h_j \mid j> m}, \end{align*}$$

where $\tilde {\alpha }_i = m - \alpha _i$ , and $\mathrm {sgn}(\alpha ) = \# \{1 \leq i \leq k \mid \alpha _i = 0\}$ (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 the set of the paths in $\operatorname {LRP}(m,n+km)^{\tilde \alpha }$ and a subset of the paths in $\operatorname {LRP}(m+k,n+k)_{\ast k}$ whose labels depend on $\alpha $ (see Proposition 4.10). This map is an adaptation of [Reference Gillespie, Gorsky and GriffinGGG25, 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 $(m+k) \times (n+k)$ .

This leads to the following definition.

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

$$\begin{align*}\alpha_i {:=}q \# \{ j \in \mathcal{D} \mid \overline{w}_j = \overline{i} \}, \end{align*}$$

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

Definition 4.9. 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 right to left in case of a tie. We say that a fall-labeling is allowable if it is a Fubini ranking, that is, $w^\ast (\pi , \overline {w})$ has allowable content. See Figure 9 for an example.

Figure 9

A path $\pi \in \operatorname {LRP}(12,8)_{\ast 4}$ with a fall-labeling $\overline {w}$ . Left to right, the vertical distances of the decorated falls from the broken diagonal are: $\frac 32, \frac 12, -\frac 12, -\frac 12$ . Therefore, the star word is $w^\ast (\pi , \overline {w}) = \overline {2}\,\overline {2}\,\overline {1}\,\overline {4}$ ; here, the label $\overline {1}$ appears before $\overline {4}$ in the star word because the step labeled $\overline {1}$ appears after the step labeled $\overline {4}$ . The fall-labeling is allowable because the content of $w^\ast (\pi , \overline {w})$ is $(1,2,0,1)$ , which is allowable.

Figure 9 Long description

Starting at the bottom left, the blue lattice path ascends in steps, alternating vertical and horizontal segments. The path begins at the grid point labeled 2, then rises vertically to 1, then horizontally right to 4, then up to 5, right to 2, up to 4, right to 2, up to 5, and finally right to 1 at the top right. Four horizontal segments are highlighted in dark red and labeled, from left to right, with barred 2 star, barred 2 star, barred 4 star, and barred 1 star. These decorated steps are positioned at increasing x-coordinates. A gray broken diagonal line runs from the lower left to the upper right, crossing the path at various points. The vertical distances from each decorated horizontal step to the diagonal are, in order, three-halves, one-half, negative one-half, and negative one-half. The numbers 2, 4, 1, 5, 2, 4, 2, 5, and 1 are written at the start of each corresponding segment along the path. The order of the barred labels in the star word is barred 2, barred 2, barred 1, barred 4, reflecting the order determined by the vertical distances and the labeling rule described.

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

Proposition 4.10. The map $\psi _0$ lifts to a bijection $\psi $ between $\operatorname {LRP}(m,n+km)^{\tilde \alpha }$ and the set

$$\begin{align*}\{ (\pi, \overline{w}) \mid \pi \in \operatorname{LRP}(m+k,n+k)_{\ast k}, \; \overline{w} \in \operatorname{FL}(\pi, \alpha) \}, \end{align*}$$

where $\overline {w}$ is obtained as in Remark 4.7.

The definitions of $\operatorname {dinv}$ , $\operatorname {tdinv}$ , 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 $\operatorname {LRP}(m, n+km)^{\tilde \alpha }$ as follows: before every horizontal step j, insert k South steps, where the first $a_j$ are nondecorated and the last $k - a_j$ are decorated (here, $a_j$ is as in Definition 4.6). 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 nondecorated South steps (naturally in bijection with $\mathcal {B}$ ); we have $\# \mathcal {D} = k$ and $\#\mathcal {S} = \#\mathcal {B} = k(m-1)$ . See Figure 10 (top right).

Figure 10

An example of the map $\psi $ in Definition 4.6 for $m=5$ , $n=2$ , $k=3$ , and $\tilde \alpha = (4,4, 4)$ . On the left, a path $\tilde \pi \in \operatorname {LRP}(m, n+km)^{\tilde \alpha }$ with its big labels (and the corresponding vertical steps $i \in \mathcal {B}$ ) highlighted in red. On the top right, the ENS representation of $\tilde \pi $ . On the center right, the ENS representation of $(\pi , \overline {w}) = \psi (\tilde \pi )$ ; it is obtained from the ENS representation of $\tilde \pi $ by pruning away the steps in $\mathcal {B} \sqcup \mathcal {S}$ . On the bottom right, the standard representation of $\pi $ . In all paths, the two squares contributing to the area are highlighted in gray, and the fall-labeling $\overline {w}$ is shown next to the decorated steps. In this example, the star word is $w^\ast (\pi , \overline {w}) = \overline {1} \, \overline {2}\, \overline {3}$ .

Figure 10 Long description

The leftmost panel shows a lattice path on a grid with vertical and horizontal steps, labeled from bottom to top with 1, 2, 3, 4, 5 and overlined 1, 2, 3. Red highlights indicate big labels and corresponding vertical steps. Two squares along the path are shaded gray. The top right panel, labeled E N S, displays the E N S representation of the path, splitting the path into three columns, each with vertical steps labeled 1, 2, 3, 4, 5 and overlined 1, 2, 3, with stars marking certain steps. The center right panel, labeled psi, shows the pruned E N S representation, where steps in sets B and S are removed, leaving only steps with fall-labels and the two gray squares. The bottom right panel, labeled E N S inverse, presents the standard representation of the resulting path, with steps labeled 1, 2, 3, 4, 5 and overlined 1, 2, 3, stars next to decorated steps, and the same two gray area squares. Arrows indicate the transformation sequence between panels.

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 10, 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 4.11. Let $\tilde \pi \in \operatorname {LRP}(m,n+km)^{\tilde \alpha }$ . For any step $i \in \mathcal {H} \sqcup \mathcal {V}$ (horizontal, or vertical with a small label), its vertical distance is preserved by $\psi $ :

$$\begin{align*}v_i(\tilde\pi) = v_i(\psi(\tilde\pi)). \end{align*}$$

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

$$\begin{align*}\{ (\pi, \overline{w}) \mid \pi \in \operatorname{LRD}(m+k,n+k)_{\ast k}, \, \overline{w} \in \operatorname{FL}(\pi, \alpha) \}. \end{align*}$$

Remark 4.12. 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 $\operatorname {tdinv}(\tilde \pi )$ if and only if it contributes to $\operatorname {tdinv}(\pi )$ . The remaining contributions to $\operatorname {tdinv}(\tilde \pi )$ are due to attacking pairs $(i, j) \in (\mathcal {V} \sqcup \mathcal {B}) \times \mathcal {B}$ . We call $\operatorname {falldinv}(\pi , \overline {w})$ the number of such contributions, so that

$$\begin{align*}\operatorname{tdinv}(\tilde\pi) = \operatorname{tdinv}(\pi) + \operatorname{falldinv}(\pi, \overline{w}). \end{align*}$$

4.3 Sign-reversing involution

Let $\pi \in \operatorname {LRP}(m+k,n+k)_{\ast k}$ , and for $\overline {w} \in \operatorname {FL}(\pi )$ , let $\mathrm {sgn}(\overline {w})$ be the sign of its content. We aim to define a sign-reversing involution on $\operatorname {FL}(\pi )$ 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 [Reference Gillespie, Gorsky and GriffinGGG25].

We have the following result.

We can lift $\varphi $ to $\operatorname {FL}(\pi )$ 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 4.15. For $\pi \in \operatorname {LRP}(m+k,n+k)_{\ast k}$ , and for $\overline {w} \in \operatorname {FL}(\pi )$ , we have

$$\begin{align*}\operatorname{falldinv}(\pi, \overline{w}) = \operatorname{falldinv}(\pi, \varphi(\overline{w})). \end{align*}$$

5.1 Computing the dinv

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

Theorem 5.1. Let $\pi \in \operatorname {LRP}(m+k,n+k)_{\ast k}$ , and let $\overline {w} \in \operatorname {FL}(\pi )$ 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

$$\begin{align*}\operatorname{dinv}(\pi) = \operatorname{dinv}(\tilde\pi). \end{align*}$$

Before proceeding with the proof, let us make some remarks and introduce some notation. First, notice that if $k=0$ , then $\psi $ is the identity and Theorem 5.1 holds, so let us assume $k> 0$ .

To lighten the notation, we denote sets referring to the path $\pi $ (and to $\overline {w}$ ) with plain letters, and sets referring to $\tilde {\pi }$ with a tilde. For example, $C_+ {:=}q C_+(\pi )$ , $\tilde {C}_- {:=}q C_-(\tilde {\pi })$ , 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 10).

Recall (cf. Definition 3.18) that

$$\begin{align*}\operatorname{cdinv}(\tilde \pi) {:=}q \# \tilde{C}_+ - \# \tilde{C}_- + \# \tilde{C}^\ast + \# \tilde{B}. \end{align*}$$

Since decorated steps of (the ENS representation of) $\tilde \pi $ do not contribute to $\operatorname {cdinv}(\tilde \pi )$ , we have $\tilde {C}^\ast = \varnothing $ ; since $k>0$ , we have $m < n + km$ and therefore $\tilde {C}_+ = \varnothing $ . Let us partition $\tilde {C}_-$ and $\tilde B$ based on the contributions of $\mathcal {V}$ (steps with small labels) and $\mathcal {B}$ (steps with big labels) as follows:

$$ \begin{align*} \tilde{C}_{\text{s}} & {:=}q \left\{ (i, j) \in \mathcal{V} \times \mathcal{H} \mid j < i \text{ and } v_j \leq v_i < v_j + \textstyle\frac{n}{m} + k - 1 \right\} \\ \tilde{C}_{\text{b}} & {:=}q \left\{ (i, j) \in \mathcal{B} \times \mathcal{H} \mid j < i \text{ and } v_j \leq v_i < v_j + \textstyle\frac{n}{m} + k - 1 \right\} \\ \tilde{B}_{\text{s}} & {:=}q \left\{ i \in \mathcal{V} \mid v_i < 0 \right\} \\ \tilde{B}_{\text{b}} & {:=}q \left\{ i \in \mathcal{B} \mid v_i < 0 \right\}. \end{align*} $$

Remark 5.2. In the previous expressions for $\tilde {C}_{\text {s}}$ and $\tilde {C}_{\text {b}}$ , the $+\,k$ term appears because the slope of the main diagonal of $\tilde \pi $ is

$$\begin{align*}\textstyle \frac{n+km}{m} = \frac{n}{m} + k. \end{align*}$$

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). See Figure 11 for an example.

Figure 11

A pair of steps $(i, j) \in \tilde {C}_{\text {s}}$ . The endpoints of $i \in \mathcal {V}$ both diagonally project onto one of the three South steps preceding $j\in \mathcal {H}$ .

Figure 11 Long description

This is a grid-based schematic. At the lower left, step i is labeled in bold. From i, two dashed diagonal lines extend upward and rightward, passing through three grid edges marked with asterisks, each representing a South step. These asterisks are distributed along the diagonal between i and j. At the lower right, step j is labeled in bold and lies horizontally. The vertical and horizontal segments are colored dark purple and blue, respectively. The grid lines are light gray, and the diagonal projection visually connects i to the three South steps before reaching j.

Partition also the contributions to $\operatorname {falldinv}(\pi , \overline {w})$ based on the attack relations as follows:

$$ \begin{align*} F_{\text{s}}^\swarrow & {:=}q \left\{ (i, j) \in \mathcal{V} \times \mathcal{B} \mid j < i \text{ and } i \rightarrow j \right\} \\ F_{\text{b}}^\swarrow & {:=}q \left\{ (i, j) \in \mathcal{B} \times \mathcal{B} \mid j < i, \; w_i < w_j, \text{ and } i \rightarrow j \right\} \\ F^\nearrow & {:=}q \left\{ (i, j) \in (\mathcal{V} \sqcup \mathcal{B}) \times \mathcal{B} \mid i < j, \; w_i < w_j, \text{ and } i \rightarrow j \right\}. \end{align*} $$

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.

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 $i' \in \mathcal {S}$ (recall that there is a natural bijection between $\mathcal {B}$ and $\mathcal {S}$ sending $i \in \mathcal {B}$ to the step $i' \in \mathcal {S}$ with the same endpoints). Note that $\pi _{\overline {1}} = \pi $ and $\pi _{\overline {k+1}} = \tilde \pi $ . Two examples are shown in Figures 12 and 13.

Figure 12

The subpaths $\pi _s$ constructed from the paths of Figure 10. The half-lines $\ell _i$ and the corresponding steps $i \in \mathcal {B}$ with $w_i = s$ are also shown. We omit the small labels.

Figure 12 Long description

From left to right, the first panel is labeled pi sub 3 bar below the grid. It shows three vertical columns of steps, each intersected by slanted half-lines labeled 3 bar, 2 bar, and 1 bar at the top. Each vertical segment is labeled from top to bottom as 3 bar, 2 bar, and 1 bar, with horizontal segments labeled 2 bar, 1 bar, and asterisk. The second panel, labeled pi sub 2 bar, has two main columns with half-lines labeled 2 bar and 1 bar at the top. Vertical segments are labeled 2 bar and 1 bar, with horizontal segments labeled 2 bar, 1 bar, and asterisk. The third panel, labeled pi sub 1 bar, has one main column with half-lines labeled 1 bar at the top. Vertical segments are labeled 1 bar, with horizontal segments labeled 2 bar, 1 bar, and asterisk. All panels use a grid background, and the subpaths are highlighted in blue, with half-lines in gray and step labels in black. The asterisk marks the bottom horizontal segment in each panel.

Figure 13

The subpaths $\pi _s$ and half-lines $\ell _i$ in another example with $m=5$ , $n=2$ , $k=3$ , and $w^\ast (\pi , \overline {w}) = \overline {1} \, \overline {2}\, \overline {3}$ . In the third picture, two half-lines start below the path, so the set $\tilde {B}_{\text {b}}$ consists of two steps with label $\overline {1}$ . In this example, part of the path (in the ENS representation) lies below the x-axis.

Figure 13 Long description

From left to right, each panel displays a square grid with a blue lattice path composed of vertical and horizontal steps. In the left panel, the path pi sub 3 is highlighted, with vertical steps labeled bar 3, bar 2, bar 1 from top to bottom, and horizontal steps labeled 2, 1, 3, 2, 1. Three diagonal half-lines extend from the lower left, intersecting the path at labeled steps. Asterisks mark steps intersected by half-lines, with bar 1, bar 2, bar 3 labels adjacent to these steps. The middle panel shows pi sub 2, with similar structure but different step arrangement and labels bar 2, bar 1, 2, 1, 2. The right panel shows pi sub 1, with the path and half-lines shifted upward, and step labels bar 1, 1, 2. In all panels, some steps and labels are below the x-axis, and the arrangement of asterisks and half-lines changes to illustrate the set tilde B sub b. All axes and grid lines are unnumbered.

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 count the number of intersections of these half-lines with certain steps of the path: 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.

Let us now state a few useful counting lemmas.

Proof. The proof strategy is to interpret all numbers in the statement as counting pairs of steps $i \in \mathcal {B}$ and $j \in \mathcal {H}$ , with $j < i$ , satisfying certain conditions. For any such pair $(i, j)$ , denote by $r_{ij}'$ the number of decorated steps $i' \in \mathcal {D}$ immediately preceding step j with $\overline {w}(i') \geq w_i$ . If the step with fall-label 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 $v_{i_*} = v_j + \frac {n}{m} + r_{ij}' - 1$ by definition of $i_*$ . Let $r_{ij}" = 1$ if $i_*$ is defined and $0$ otherwise.

For a fixed step $i \in \mathcal {B}$ , let us count the intersections of the half-line $\ell _i$ with the steps in $\mathcal {H} \sqcup \mathcal {D} \sqcup \mathcal {S}$ belonging to $\pi _{w_i}$ . This is equal to the number of steps $j \in \mathcal {H}$ with $j < i$ such that $\ell _i$ intersects step j or one of the steps in $\mathcal {D} \sqcup \mathcal {S}$ immediately before step j belonging to $\pi _{w_i}$ (see Figure 14). For a given $j \in \mathcal {H}$ , this happens if and only if

(5.1) $$ \begin{align} v_j \leq v_i < v_j + \textstyle\frac{n}{m} + r_{ij}' + w_i - 1. \end{align} $$

Here we treat $w_i$ as a positive integer, even though technically $w_i \in \overline {\mathbb {Z}}_+$ .

Figure 14

Examples of pairs $(i, j) \in \mathcal {B} \times \mathcal {H}$ satisfying (5.1) in the proof of Lemma 5.5. The two cases differ by the label $w_i$ , which is $\overline {4}$ on the left and $\overline {2}$ on the right; consequently, the path $\pi _{w_i}$ on the right contains fewer steps. We have $r_{ij}' = 1$ on the left and $r_{ij}' = 2$ on the right.

Figure 14 Long description

The left panel shows a vertical path labeled j at the base, with segments marked overline 2, asterisk, overline 5, asterisk, and a diagonal segment labeled l sub i connecting to i at the top. The diagonal is labeled overline 1, overline 3, overline 4. Braces on the left indicate w sub i minus 1 and r prime sub i j equals 1. The right panel is similar but the diagonal segment is shorter, labeled overline 1, overline 2, and connects to i at the top. The vertical path is labeled overline 2, asterisk, overline 5, asterisk, with r prime sub i j equals 2. Both panels use color to distinguish segments and labels.

Next, let us rewrite the term $\# F_{\text {b}}^\swarrow $ . For $i \in \mathcal {B}$ and $j \in \mathcal {H}$ with $j < i$ , there is (exactly) one pair $(i, i') \in F_{\text {b}}^\swarrow $ , with $i'$ appearing above the left-most endpoint of j, if and only if

(5.2) $$ \begin{align} v_j + \textstyle\frac{n}{m} + r_{ij}' + w_i - 1 - r_{ij}" \leq v_i < v_j + \textstyle\frac{n}{m} + k - 1; \end{align} $$

this is exemplified in Figure 15.

Figure 15

Examples of pairs $(i, j) \in \mathcal {B} \times \mathcal {H}$ satisfying (5.2) in the proof of Lemma 5.5. The dashed gray line indicates the attacking relation: step i attacks a step $i'$ above the left-most endpoint of j. Note that $r_{ij}" = 0$ on the left ( $i_*$ is not defined) and $r_{ij}" = 1$ on the right ( $i_*$ is defined).

Figure 15 Long description

The left panel shows a vertical stack anchored at the bottom by a horizontal segment labeled j in blue. Above j, two segments are marked with asterisks and labeled 2 and 5 in black. Above these, four segments are labeled 1, 3, 4, and 6 in red, increasing upward. A dashed gray line extends from the topmost red segment labeled 4 to a red index i on the right. Braces on the left indicate k, 1 minus r double prime sub i j, w sub i minus 1, and r prime sub i j, each spanning different portions of the stack. The right panel is similar but includes an additional index i star between the red and black segments, and the dashed line connects from the segment labeled 2 to i. The braces on the right indicate k, w sub i minus 1, and r prime sub i j. The difference between panels is the presence of i star and the value of r double prime sub i j, which is zero on the left and one on the right.

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

(5.3) $$ \begin{align} v_j + \textstyle\frac{n}{m} + r_{ij}' + w_i - 1 - r_{ij}" \leq v_i < v_j + \textstyle\frac{n}{m} + r_{ij}' + w_i - 1 \end{align} $$

(note that this condition is vacuous if $r_{ij}" = 0$ ). See Figure 16.

Figure 16

Example of a pair $(i, j) \in \mathcal {B} \times \mathcal {H}$ satisfying (5.3) in the proof of Lemma 5.5. The dashed gray line indicates that, in order to have $(i_*, j') \in C^*$ , the top-most endpoint of $j'$ must diagonally project onto $i_*$ .

Figure 16 Long description

At the center left, a vertical red bar labeled i is marked with overlined numbers 6, 4, 3, and 1 from top to bottom. To the right of i, a shorter vertical bar labeled overline 2 i is present. At the base of i, a horizontal purple bar labeled j extends rightward, annotated with 2 star and 5 star from top to bottom. The intersection point at the base is labeled i sub star. From i sub star, a dashed gray line projects diagonally upward and rightward to a vertical bar labeled overline 3 star j prime. To the right of this, a curly brace encloses the vertical span, labeled w sub i minus 1.

The term $\# \tilde C_{\text {b}}$ is already defined in terms of pairs $(i, j) \in \mathcal {B} \times \mathcal {H}$ with $j < i$ ; such a pair contributes to $\# \tilde C_{\text {b}}$ if and only if

(5.4) $$ \begin{align} v_j \leq v_i < v_j + \textstyle\frac{n}{m} + k - 1. \end{align} $$

We conclude by observing that a pair $(i, j)$ satisfies (5.1) or (5.2) if and only if (5.4) holds; it satisfies both (5.1) and (5.2) if and only if (5.3) holds.

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

$$\begin{align*}F_{\text{s}}' = \left\{ (i, j) \in \mathcal{V} \times \mathcal{H} \mid j < i \text{ and } v_j + \textstyle\frac{n}{m} + r_j - 1 \leq v_i < v_j + \textstyle\frac{n}{m} + k - 1 \right\}, \end{align*}$$

and clearly $\# F_{\text {s}}' = \# F_{\text {s}}^\swarrow $ . Now, $C_+ \sqcup \tilde {C}_{\text {s}} = C_- \sqcup F_{\text {s}}'$ (the sets are disjoint) as they are both equal to

$$\begin{align*}\left\{ (i, j) \in \mathcal{V} \times \mathcal{H} \mid j < i \text{ and } v_j + \min\left(0, \, \textstyle\frac{n}{m} + r_j - 1 \right) \leq v_i < v_j + \textstyle\frac{n}{m} + k - 1 \right\}. \end{align*}$$

The result follows.

We can now proceed with the proof of Theorem 5.1.

Proof of Theorem 5.1

By definition, we have $\operatorname {dinv}(\pi ) = \operatorname {tdinv}(\pi ) + \operatorname {cdinv}(\pi )$ and $\operatorname {dinv}(\tilde \pi ) = \operatorname {tdinv}(\tilde \pi ) + \operatorname {cdinv}(\tilde \pi )$ . By Remark 4.12, we have $\operatorname {tdinv}(\tilde \pi ) = \operatorname {tdinv}(\pi ) + \operatorname {falldinv}(\pi , \overline {w})$ , so it remains to prove the following:

$$\begin{align*}\operatorname{cdinv}(\pi) = \operatorname{cdinv}(\tilde\pi) + \operatorname{falldinv}(\pi, \overline{w}). \end{align*}$$

Using the previously shown partitionings, we can rewrite the statement as

(5.5) $$ \begin{align} \# C_+ - \# C_- + \# C^* + \# B = - \# \tilde{C}_{\text{s}} - \# \tilde{C}_{\text{b}} + \# \tilde B_{\text{s}} + \# \tilde B_{\text{b}} + \# F_{\text{s}}^\swarrow + \# F_{\text{b}}^\swarrow + \# F^\nearrow. \end{align} $$

By construction, we have $B = \tilde {B}_{\text {s}}$ . By Lemma 5.6, we have $\# C_+ - \# C_- = \# F_{\text {s}}^\swarrow - \# \tilde {C}_{\text {s}}$ . Therefore, we can cancel out the corresponding summands from (5.5) and reduce the statement to

$$\begin{align*}(\# C^\ast + \# \tilde{C}_{\text{b}} - \# F_{\text{b}}^\swarrow) - \# F^\nearrow = \# \tilde B_{\text{b}}. \end{align*}$$

By Lemma 5.4, the term $\# F^\nearrow $ counts the intersections of the half-lines $\ell _i$ with the steps in $\mathcal {V} \sqcup \mathcal {B}$ belonging to $\pi _{w_i}$ ; by Lemma 5.5, the term $\#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 $\pi _{w_i}$ .

Now, fix $i \in \mathcal {B}$ . If we traverse $\ell _i$ in the top-right direction, every time $\ell _i$ intersects a step in $\mathcal {H} \sqcup \mathcal {D} \sqcup \mathcal {S}$ it goes from weakly below the path $\pi _{w_i}$ to strictly above the path, and every time it intersects a step in $\mathcal {V} \sqcup \mathcal {B}$ it does the opposite. Since $\ell _i$ ends weakly above the path, these numbers are the same if $\ell _i$ starts weakly above the path, and they are off by one if $\ell _i$ starts strictly below the path, in which case $i \in \tilde B_{\text {b}}$ . Summing the contributions over all steps $i \in \mathcal {B}$ , the result follows.

5.2 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 5.7 (Rational shuffle theorem [Reference MellitMel21])

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

$$\begin{align*}e_{m,n} = \sum_{\pi \in \operatorname{LRD}(m,n)} q^{\operatorname{dinv}(\pi)} t^{\operatorname{area}(\pi)} x^\pi. \end{align*}$$

Applying the skewing operator $s_{(m-1)^k}^\perp $ to the combinatorial interpretation of $e_{m,n+km}$ given by Theorem 5.7, and using Proposition 4.5, we obtain

$$\begin{align*}s_{(m-1)^k}^\perp e_{m,n+km} = \sum_{\alpha \text{ allowable}} (-1)^{\mathrm{sgn}(\alpha)} \sum_{\tilde\pi \in \operatorname{LRD}(m,n+km)^{\tilde\alpha}} q^{\operatorname{dinv}(\tilde\pi)} t^{\operatorname{area}(\tilde\pi)} x^{\tilde\pi}. \end{align*}$$

Combining this with Proposition 4.10, we get

$$\begin{align*}s_{(m-1)^k}^\perp e_{m,n+km} = \sum_{\pi \in \operatorname{LRD}(m+k,n+k)_{\ast k}} \sum_{\overline{w} \in \operatorname{FL}(\pi)} (-1)^{\mathrm{sgn}(\overline{w})} q^{\operatorname{dinv}(\psi^{-1}(\pi, \overline{w}))} t^{\operatorname{area}(\psi^{-1}(\pi, \overline{w}))} x^\pi. \end{align*}$$

By Proposition 4.15 and Theorem 5.1, this equality becomes

$$\begin{align*}s_{(m-1)^k}^\perp e_{m,n+km} = \sum_{\pi \in \operatorname{LRD}(m+k,n+k)_{\ast k}} q^{\operatorname{dinv}(\pi)} t^{\operatorname{area}(\pi)} x^\pi, \end{align*}$$

which is exactly Theorem 1.1, as desired.

Let us recall the statement of the rectangular shuffle conjecture.

Conjecture 5.8 ([Reference Iraci, Pagaria, Paolini and Vanden WyngaerdIPPVW23, Conjecture 4.2])

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

$$\begin{align*}\frac{[m]_q}{[d]_q} p_{m,n} = \sum_{\pi \in \operatorname{LRP}(m,n)} q^{\operatorname{dinv}(\pi)} t^{\operatorname{area}(\pi)} x^\pi. \end{align*}$$

Using the fact that $\gcd (m,n+km) = \gcd (m,n)$ , the previous argument also proves Theorem 1.2. In particular, due to [Reference Iraci, Pagaria, Paolini and Vanden WyngaerdIPPVW23, Theorem 6.1], the equality of Theorem 1.2 holds when $d=1$ .

6.1 $D_\alpha $ operators and e-positivity

Let $\alpha \vDash n$ be defined by $\alpha _i {:=}q \left \lceil \frac {ni}{m} \right \rceil - \left \lceil \frac {n(i-1)}{m} \right \rceil $ , and let $x^\alpha {:=}q \prod _{i} x_i^{\alpha _i}$ . Recall that, in the notation of [Reference Blasiak, Haiman, Morse, Pun and SeelingerBHM+23b] (cf. [Reference Blasiak, Haiman, Morse, Pun and SeelingerBHM+23a, Reference Gillespie, Gorsky and GriffinGGG25]), we have the identity

$$\begin{align*}e_{m,n} = D_\alpha(1) = \omega \left( H_{q,t}^m \left( \frac{x^\alpha}{\prod_i (1-qtx_i/x_{i+1})} \right)_{\mathrm{pol}} \right). \end{align*}$$

Then, by [Reference Gillespie, Gorsky and GriffinGGG25, Lemma 3.12], we can deduce the following identity.

Proposition 6.1. For $m,n,k \in \mathbb {N}$ with $m> 0$ , and $\alpha $ as above, we have

$$\begin{align*}s_{(m-1)^k}^{\perp} e_{m,n+km} = \omega \left(H_{q,t}^m \left( \frac{x^\alpha h_k(x_1, \dots, x_m)}{\prod_i (1-qtx_i/x_{i+1})} \right)_{\mathrm{pol}} \right). \end{align*}$$

Since the operator $ H_{q,t}^m \left ( \cdot \right )_{\mathrm {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 $x_i$ .

We can naturally split the generating function of $\operatorname {LRD}(m+k,n+k)_{\ast k}$ 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 6.3. 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

$$\begin{align*}D_{\alpha+\beta}(1) \left\vert_{q=1} \right.= \sum_{\substack{\pi \in \operatorname{LRD}(m+k,n+k)_{\ast k} \\ \beta(\pi) = \beta}} t^{\operatorname{area}(\pi)} x^{\pi}\end{align*}$$

Theorem 6.3 is essentially a reformulation of [Reference Blasiak, Haiman, Morse, Pun and SeelingerBHM+23b, Equation (148)], which states

$$\begin{align*}D_\gamma(1) = \sum_{\lambda} t^{a(\lambda)} e_{b(\lambda)}, \end{align*}$$

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 6.3, 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 6.3 has the same combinatorial flavor as [Reference Hicks and RomeroHR18, Theorem 1.1] (cf. [Reference Iraci and RomeroIR24, Theorem 8.6]), as they both split a positive generating function into nonpositive pieces that become positive and combinatorial when specializing one variable to $1$ .

6.2 Theta operators

Another attempt at finding a decorated extension of the rational shuffle theorem is [Reference Iraci, Pagaria, Paolini and Vanden WyngaerdIPPVW23, Conjecture 4.4], which states the following.

Conjecture 6.4. For any $m, n \in \mathbb {N}$ , we have

$$\begin{align*}\left. \Theta_{e_{k}} e_{m,n} \right\rvert_{q=1} = \sum_{\pi \in \operatorname{LRD}(m+k,n+k)^{\ast k}} t^{\operatorname{area}(\pi)} x^\pi, \end{align*}$$

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

The main difference from Theorem 1.1 is that the decorations are on the rises rather than the falls. The problem of finding a $\operatorname {dinv}$ 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 nonlabeled 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 6.5. For $m, n, k, d \in \mathbb {N}$ with $m> 0$ and $d \leq \min \{m+k, n+k\}$ , we have

$$\begin{align*}\langle e_{m,n+km}, s_{(m-1)^k} h_d e_{n+k-d} {\rangle} = \langle \Theta_{e_k} e_{n,m}, h_d e_{m+k-d} {\rangle}. \end{align*}$$

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

6.3 The Delta square conjecture

In [Reference D’Adderio, Iraci and Vanden WyngaerdDIVW19], the authors conjecture a decorated version of the square paths theorem, in terms of rise-decorated square paths. We refer to [Reference D’Adderio, Iraci and Vanden WyngaerdDIVW19, Reference Iraci and Vanden WyngaerdIVW21] for all missing definitions. The statement is the following.

Conjecture 6.6 ([Reference D’Adderio, Iraci and Vanden WyngaerdDIVW19, Conjecture 3.12])

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

$$\begin{align*}\frac{[n]_t}{[n+k]_t} \Delta_{e_{n}} \omega(p_{n+k}) = \sum_{\pi \in \operatorname{LRP}(n+k,n+k)^{\ast k}} q^{\operatorname{dinv}(\pi)} t^{\operatorname{area}(\pi)} x^\pi, \end{align*}$$

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

Notice the extra factor $[n]_t/[n+k]_t$ in the expression. In [Reference Iraci and Vanden WyngaerdIVW21], the authors partly address this issue. Via the identity

$$\begin{align*}\frac{[n]_t}{[n+k]_t} \Delta_{e_{n}} \omega(p_{n+k}) = \frac{[n+k]_q}{[n]_q} \Theta_{e_k} \nabla \omega(p_n) \end{align*}$$

and the $q,t$ -reversed version

$$\begin{align*}\frac{[n]_q}{[n+k]_q} \Delta_{e_{n}} \omega(p_{n+k}) = \frac{[n+k]_t}{[n]_t} \Theta_{e_k} \nabla \omega(p_n), \end{align*}$$

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 it is strictly smaller than the label of the vertical step immediately after it. These decorations first appeared in the valley version of the Delta conjecture [Reference Haglund, Remmel and WilsonHRW18]. The authors give the following two conjectures.

Conjecture 6.7 ([Reference Iraci and Vanden WyngaerdIVW21, Conjecture 5])

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

$$\begin{align*}\frac{[n]_t}{[n+k]_t} \Theta_{e_k} \nabla \omega(p_n) = \sum_{\pi \in \operatorname{LRP}(n+k, n+k)^{\bullet k}} q^{\operatorname{dinv}(\pi)} t^{\operatorname{area}(\pi)} x^\pi, \end{align*}$$

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

Conjecture 6.8 ([Reference Iraci and Vanden WyngaerdIVW21, Conjecture 8])

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

$$\begin{align*}\Theta_{e_k} \nabla \omega(p_n) = \sum_{\pi \in \operatorname{LRP}'(n+k, n+k)^{\bullet k}} q^{\operatorname{dinv}(\pi)} t^{\operatorname{area}(\pi)} x^\pi, \end{align*}$$

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

Notice that Conjecture 6.8 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 6.9 (Delta square conjecture, fall version)

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

$$\begin{align*}\Theta_{e_k} \nabla \omega(p_n) = \sum_{\pi \in \operatorname{LRP}(n+k, n+k)_{\ast k}} q^{\operatorname{dinv}(\pi)} t^{\operatorname{area}(\pi)} x^\pi. \end{align*}$$

Since we already have a formula for the right-hand side of Conjecture 6.9, we claim the following.

Conjecture 6.10. For $n, k \in \mathbb {N}$ , we have

$$\begin{align*}\Theta_{e_k} \nabla \omega(p_n) = s_{(n-1)^k}^\perp p_{n,n(k+1)}. \end{align*}$$