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\title{Pushing our way from the valley Delta to the generalised valley Delta}
\author[\initial{A.} Iraci]{\firstname{Alessandro} \lastname{Iraci}}
\address{Université du Québec à Montréal\\
Laboratoire d'Algèbre, de Combinatoire et d'Informatique Mathématique (LACIM)\\
201 Av. du Président-Kennedy\\
Montréal\\
QC H2X 3Y7 (Canada)}
\email{iraci.alessandro@uqam.ca}
\author[\initial{A.} Vanden Wyngaerd]{\firstname{Anna} \lastname{Vanden Wyngaerd}}
\address{Université de Paris\\
Institut de Recherche en Informatique Fondamentale (IRIF) \\
Bâtiment Sophie Germain, Case courrier 7014\\
8 Place Aurélie Nemours\\
Paris\\
75205 Cedex 13 (France)}
\email{avw@irif.fr}
\keywords{Delta conjecture, Theta operators, Macdonald polynomials}
\subjclass{05E05}
\datereceived{2021-10-25}
\daterevised{2022-03-22}
\dateaccepted{2022-03-23}
\datepublished{2022-09-08}
\begin{document}
\begin{abstract}
In [Haglund, Remmel, Wilson 2018] the authors state two versions of the so-called Delta conjecture, the rise version and the valley version. Of the former, they also give a more general statement in which zero labels are also allowed. In [Qiu, Wilson 2020], the corresponding generalisation of the valley version is also formulated.
In [D'Adderio, Iraci, Vanden Wyngaerd 2020], the authors use a pushing algorithm to prove the generalised version of the shuffle theorem. An extension of that argument is used in [Iraci, Vanden Wyngaerd 2020] to formulate a valley version of the (generalised) Delta square conjecture, and to suggest a symmetric function identity later stated and proved in [D'Adderio, Romero 2020].
In this paper, we use the pushing algorithm together with the aforementioned symmetric function identity in order to prove that the valley version of the Delta conjecture implies the valley version of the generalised Delta conjecture, which means that they are actually equivalent.
Combining this with the results in [Iraci, Vanden Wyngaerd 2020], we prove that the valley version of the Delta conjecture also implies the corresponding generalised Delta square conjecture.
\end{abstract}
\maketitle
\section{Introduction}
In \cite{Haglund-Remmel-Wilson-2018}, Haglund, Remmel and Wilson conjectured a combinatorial formula for $\Delta_{e_{n-k-1}}'e_n$ in terms of decorated labelled Dyck paths, which they called the \emph{Delta conjecture}, after the so-called Delta operators $\Delta_f'$ introduced by Bergeron, Garsia, Haiman, and Tesler \cite{Bergeron-Garsia-Haiman-Tesler-Positivity-1999} for any symmetric function $f$.
There are two versions of the conjecture, referred to as the \emph{rise} and the \emph{valley} version.
The case $k=0$ of the Delta conjecture is the famous \emph{shuffle theorem} which was proved by Carlsson and Mellit \cite{Carlsson-Mellit-ShuffleConj-2018}, using the \emph{compositional refinement} formulated in \cite{Haglund-Morse-Zabrocki-2012}; the shuffle theorem, thanks to the famous \emph{$n!$ theorem} of Haiman \cite{Haiman-nfactorial-2001}, gives a combinatorial formula for the Frobenius characteristic of the $\mathfrak{S}_n$-module of diagonal harmonics studied by Garsia and Haiman.
According to a conjecture by Zabrocki \cite{Zabrocki-Delta-Module-2019}, both versions of the Delta conjecture provide a conjectural combinatorial formula for the Frobenius characteristic of the so-called ``super diagonal coinvariants''; Haglund and Sergel \cite{Haglund-Sergel-2021} (after the breakthrough of Carlsson and Oblomkov \cite{Carlsson-Oblomkov-2019}) later provided a (conjectural) explicit basis of this $\mathfrak{S}_n$ module which would explain the combinatorial formula for its Hilbert series coming from the valley Delta (and not from the rise Delta).
A compositional refinement of the rise version of the Delta conjecture was announced in \cite{DAdderio-Iraci-VandenWyngaerd-Theta-2021} and proved in \cite{DAdderio-Mellit-Compositional-Delta-2020}, with both breakthroughs relying heavily on the novel \emph{Theta operators} introduced in \cite{DAdderio-Iraci-VandenWyngaerd-Theta-2021}.
The \emph{generalised Delta conjecture} is a combinatorial formula for $\Delta_{h_m}\Delta_{e_{n-k-1}}'e_n$ in terms of decorated partially labelled Dyck paths (the rise version first appeared in \cite{Haglund-Remmel-Wilson-2018}, the valley version in \cite{Qiu-Wilson-2020}); the rise version has been recently proved in \cite{Blasiak-Haiman-Morse-Pun-Seeling-Extended-Delta-2021}, while the valley version of the Delta conjecture (as its generalisation) remains an open problem today.
Using the Theta operators, we conjectured a \emph{touching refinement} (where the number of times the Dyck path returns to the main diagonal is specified) of the valley version of the (generalised) Delta conjecture \cite{Iraci-VandenWyngaerd-Valley-Square-2021}; in this paper, we prove that the touching refinement of the valley version of the Delta conjecture implies the touching refinement of the valley version of the generalised Delta conjecture.
Our proof will rely on a new symmetric function identity proved in \cite{DAdderio-Romero-Theta-Identities-2020}, which was suggested by a combinatorial argument we call the \emph{pushing algorithm} first described in \cite{DAdderio-Iraci-VandenWyngaerd-Theta-2021} for paths with no decorations and then extended in \cite{Iraci-VandenWyngaerd-Valley-Square-2021} to paths with decorated contractible valleys; combining this result with the results in \cite{Iraci-VandenWyngaerd-Valley-Square-2021}, we obtain that, if the valley version of the Delta conjecture is true, then the valley version of the generalised Delta square conjecture is also true. Thus, the main conjecture implies three other statements: the generalised version, the square version, and the generalised square version.
Note that, despite the rise version of the generalised Delta conjecture being proved, it does not feature an analogue of the results in \cite{Iraci-VandenWyngaerd-Valley-Square-2021}, and so the rise version of the Delta square conjecture \cite{DAdderio-Iraci-VandenWyngaerd-DeltaSquare-2019} remains open.
\section{Symmetric functions}
For all the undefined notations and the unproven identities, we refer to \cite[Section~1]{DAdderio-Iraci-VandenWyngaerd-TheBible-2019}, where definitions, proofs and/or references can be found.
We denote by $\Lambda$ the graded algebra of symmetric functions with coefficients in $\mathbb{Q}(q,t)$, and by $\<\, , \ra$ the \emph{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 a partition $\mu \vdash n$, we denote by \[ \HH_\mu \coloneqq \HH_\mu[X] = \HH_\mu[X; q,t] = \sum_{\lambda \vdash n} \widetilde{K}_{\lambda \mu}(q,t) s_{\lambda} \] the \emph{(modified) Macdonald polynomials}, where \[ \widetilde{K}_{\lambda \mu} \coloneqq \widetilde{K}_{\lambda \mu}(q,t) = K_{\lambda \mu}(q,1/t) t^{n(\mu)} \] are the \emph{(modified) Kostka coefficients} (see \cite[Chapter~2]{Haglund-Book-2008} for more details).
Macdonald polynomials form a basis of the ring of symmetric functions $\Lambda$. This is a modification of the basis introduced by Macdonald \cite{Macdonald-Book-1995}.
If we identify the partition $\mu$ with its Ferrer diagram, \ie 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 \emph{arm}, \emph{leg}, \emph{co-arm} and \emph{co-leg} (denoted respectively as $a_\mu(c), l_\mu(c), a'_\mu(c), l'_\mu(c)$) as the number of cells in $\mu$ that are respectively strictly to the right, above, to the left and below $c$ in $\mu$ (in French notation).
Let $M \coloneqq (1-q)(1-t)$. For every partition $\mu$, we define the following constants:
\[
B_{\mu} \coloneqq B_{\mu}(q,t) = \sum_{c \in \mu} q^{a_{\mu}'(c)} t^{l_{\mu}'(c)}, \qquad \qquad
\Pi_{\mu} \coloneqq \Pi_{\mu}(q,t) = \prod_{c \in \mu / (1)} (1-q^{a_{\mu}'(c)} t^{l_{\mu}'(c)}). \]
We will make extensive use of the \emph{plethystic notation} (cf.\ \cite[Chapter~1]{Haglund-Book-2008}).
We need to introduce several linear operators on $\Lambda$.
\begin{defi}[{\cite[(3.11)]{Bergeron-Garsia-ScienceFiction-1999}}]
\label{def:nabla}
We define the linear operator $\nabla \colon \Lambda \rightarrow \Lambda$ on the eigenbasis of Macdonald polynomials as \[ \nabla \HH_\mu = e_{\lvert \mu \rvert}[B_\mu] \HH_\mu. \]
\end{defi}
\begin{defi}
\label{def:pi}
We define the linear operator $\mathbf{\Pi} \colon \Lambda \rightarrow \Lambda$ on the eigenbasis of Macdonald polynomials as \[ \mathbf{\Pi} \HH_\mu = \Pi_\mu \HH_\mu \] where we conventionally set $\Pi_{\varnothing} \coloneqq 1$.
\end{defi}
\begin{defi}
\label{def:delta}
For $f \in \Lambda$, we define the linear operators $\Delta_f, \Delta'_f \colon \Lambda \rightarrow \Lambda$ on the eigenbasis of Macdonald polynomials as \[ \Delta_f \HH_\mu = f[B_\mu] \HH_\mu, \qquad \qquad \Delta'_f \HH_\mu = f[B_\mu-1] \HH_\mu. \]
\end{defi}
Observe that on the vector space of symmetric functions homogeneous of degree $n$, denoted by $\Lambda^{(n)}$, the operator $\nabla$ equals $\Delta_{e_n}$ and $\Delta'_{e_{n-1}}$.
We also introduce the Theta operators, first defined in \cite{DAdderio-Iraci-VandenWyngaerd-Theta-2021}
\begin{defi}
\label{def:theta}
For any symmetric function $f\in \Lambda^{(n)}$ we introduce the following \emph{Theta operators} on $\Lambda$: for every $F \in \Lambda^{(m)}$ we set
\begin{equation*}
\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.,
\end{equation*}
and we extend by linearity the definition to any $f,F\in \Lambda$.
\end{defi}
It is clear that $\Theta_f$ is linear, and moreover, if $f$ is homogeneous of degree $k$, then so is $\Theta_f$, \ie \[\Theta_f \Lambda^{(n)} \subseteq \Lambda^{(n+k)} \qquad \text{ for } f \in \Lambda^{(k)}. \]
It is convenient to introduce the so-called $q$-notation. In general, a $q$-analogue of an expression is a generalisation involving a parameter $q$ that reduces to the original one for $q \rightarrow 1$.
\begin{defi}
For a natural number $n \in \mathbb{N}$, we define its $q$-analogue as \[ [n]_q \coloneqq \frac{1-q^n}{1-q} = 1 + q + q^2 + \dots + q^{n-1}. \]
\end{defi}
Given this definition, one can define the $q$-factorial and the $q$-binomial as follows.
\begin{defi}
We define \[ [n]_q! \coloneqq \prod_{k=1}^{n} [k]_q \quad \text{and} \quad \qbinom{n}{k}_q \coloneqq \frac{[n]_q!}{[k]_q![n-k]_q!}. \]
\end{defi}
\begin{defi}
For $x$ any variable and $n \in \N \cup \{ \infty \}$, we define the \emph{$q$-Pochhammer symbol} as \[ (x;q)_n \coloneqq \prod_{k=0}^{n-1} (1-xq^k) = (1-x) (1-xq) (1-xq^2) \cdots (1-xq^{n-1}). \]
\end{defi}
We can now introduce yet another family of symmetric functions.
\begin{defi}
\label{def:Enk}
For $0 \leq k \leq n$, we define the symmetric function $E_{n,k}$ by the expansion \[ e_n \left[ X \frac{1-z}{1-q} \right] = \sum_{k=0}^n \frac{(z;q)_k}{(q;q)_k} E_{n,k}. \]
\end{defi}
Notice that $E_{n,0} = \delta_{n,0}$. Setting $z=q^j$ we get \[ e_n \left[ X \frac{1-q^j}{1-q} \right] = \sum_{k=0}^n \frac{(q^j;q)_k}{(q;q)_k} E_{n,k} = \sum_{k=0}^n \qbinom{k+j-1}{k}_q E_{n,k} \] and in particular, for $j=1$, we get \[ e_n = E_{n,0} + E_{n,1} + E_{n,2} + \cdots + E_{n,n}, \] so these symmetric functions split $e_n$, in some sense.
The Theta operators will be useful to restate the Delta conjectures in a new fashion, thanks to the following results.
\begin{theorem}[{\cite[Theorem~3.1]{DAdderio-Iraci-VandenWyngaerd-Theta-2021}}]
\label{thm:theta-en}
\[ \Theta_{e_k} \nabla e_{n-k} = \Delta_{e_{n-k-1}}' e_n \]
\end{theorem}
The key symmetric function identity on which our proof relies is the following. We first formulated this identity by studying the combinatorics. Its proof is due to D'Adderio and Romero.
\begin{theorem}[{\cite[Corollary~9.2]{DAdderio-Romero-Theta-Identities-2020}}]
\label{thm:sf-identity}
Given $m,n,k,r \in \N$, we have
\begin{align}
\label{eq:pushing}
h_m^\perp & \Theta_{e_k} \nabla E_{n-k,r} \\ & = \sum_{p=0}^m t^{m-p} \sum_{i=0}^p q^{\binom{i}{2}} \qbinom{r-p+i}{i}_q \qbinom{r}{p-i}_q \Delta_{h_{m-p}} \Theta_{e_{k-i}} \nabla E_{n-m-(k-i), r-p+i}.\nonumber
\end{align}
\end{theorem}
With respect to the original source \cite[Corollary~9.2]{DAdderio-Romero-Theta-Identities-2020}, we applied the change of variables $j \mapsto m, m \mapsto k, p \mapsto n-k-r, k \mapsto r, s \mapsto p, r \mapsto p-i$ in order to make it easier to interpret combinatorially and more consistent with the notation used in other papers.
\section{Combinatorial definitions}
\begin{defi}
A \emph{Dyck path} of size $n$ is a lattice path going from $(0,0)$ to $(n,n)$ consisting of east or north unit steps, always staying weakly above the line $x=y$, called the \emph{main diagonal}. The set of such paths is denoted by $\D(n)$.
\end{defi}
\begin{defi}
Let $\pi$ be a Dyck path of size $n$. We define its \emph{area word} to be the sequence of integers $a(\pi) = (a_1(\pi), a_2(\pi), \cdots, a_n(\pi))$ such that the $i$-th vertical step of the path starts from the diagonal $y=x+a_i(\pi)$.
\end{defi}
For example the path in Figure~\ref{fig:labelled-dyck-path} has area word $01101211$.
\begin{defi}
A \emph{partial labelling} of a Dyck path $\pi$ of size $n$ is an element $w \in \mathbb N^n$ such that
\begin{itemize}
\item if $a_i(\pi) > a_{i-1}(\pi)$, then $w_i > w_{i-1}$,
\item $w_1 > 0$.
\end{itemize}
\ie if we label the $i$-th vertical step of $\pi$ with $w_i$, then the labels appearing in each column of $\pi$ are strictly increasing from bottom to top, with the additional restriction that the first label cannot be a $0$.
We omit the word \emph{partial} if the labelling is composed of strictly positive labels only.
\end{defi}
\begin{defi}
A \emph{(partially) labelled Dyck path} is a pair $(\pi, w)$ where $\pi$ is a Dyck path and $w$ is a (partial) labelling of $\pi$. We denote by $\LD(m,n)$ the set of labelled Dyck paths of size $m+n$ with exactly $n$ positive labels, and thus exactly $m$ labels equal to $0$.
\end{defi}
Now we want to extend our sets by introducing some decorations.
\begin{defi}
\label{def:valley}
The \emph{contractible valleys} of a labelled Dyck path $\pi$ are the indices $2 \leq i \leq n$ such that one either $a_i(\pi) < a_{i-1}(\pi)$, or $a_i(\pi) = a_{i-1}(\pi)$ and $w_i > w_{i-1}$.
We define \[ v(\pi, w) \coloneqq \{1 \leq i \leq n \mid i \text{ is a contractible valley} \}, \] corresponding to the set of vertical steps that are directly preceded by a horizontal step and, if we were to remove that horizontal step and move it after the vertical step, we would still get a Dyck path with a valid labelling.
\end{defi}
\begin{defi}
A \emph{valley-decorated (partially) labelled Dyck path} is a triple $(\pi, w, dv)$ where $(\pi, w)$ is a (partially) labelled Dyck path and $dv \subseteq v(\pi, w)$ is a subset of the contractible valleys that we decorate.
\end{defi}
We denote by $\LD(m,n)^{\bullet k}$ the set of partially labelled valley-decorated Dyck paths of size $m+n$ with $n$ positive labels and $k$ decorated contractible valleys.
Finally, we sometimes omit writing $m$ or $k$ when they are equal to $0$. Notice that, because of the restrictions we have on the labelling and the decorations, the only path with $n=0$ is the empty path, for which also $m=0$ and $k=0$.
\begin{figure}[!ht]
\centering
\begin{tikzpicture}[scale=.6]
\draw[step=1.0, gray!60, thin] (0,0) grid (8,8);
\begin{scope}
\clip (0,0) rectangle (8,8);
\draw[gray!60, thin] (0,0) -- (8,8);
\end{scope}
\draw[blue!60, line width=2pt] (0,0) -- (0,1) -- (0,2) -- (1,2) -- (1,3) -- (2,3) -- (3,3) -- (3,4) -- (3,5) -- (3,6) -- (4,6) -- (5,6) -- (5,7) -- (6,7) -- (6,8) -- (7,8) -- (8,8);
\draw (0.5,0.5) circle (0.4cm) node {$1$};
\draw (0.5,1.5) circle (0.4cm) node {$2$};
\draw (1.5,2.5) circle (0.4cm) node {$4$};
\draw (3.5,3.5) circle (0.4cm) node {$0$};
\draw (3.5,4.5) circle (0.4cm) node {$5$};
\draw (3.5,5.5) circle (0.4cm) node {$6$};
\draw (5.5,6.5) circle (0.4cm) node {$0$};
\draw (6.5,7.5) circle (0.4cm) node {$3$};
\draw (0.5,2.5) node {$\bullet$};
\draw (4.5,6.5) node {$\bullet$};
\end{tikzpicture}
\caption{Example of an element in $\LD(2,6)^{\bullet 2}$.}
\label{fig:labelled-dyck-path}
\end{figure}
\begin{defi}
Let $w$ be a labelling of Dyck path of size $n$. We define $x^w \coloneqq \prod_{i=1}^{n} x_{w_i} \rvert_{x_0 = 1}$. For $P \coloneqq (\pi,w,dv)\in \LD(m,n)^{\bullet k}$ we define $x^P\coloneqq x^w$.
\end{defi}
The fact that we set $x_0 = 1$ explains the use of the expression \emph{partially labelled}, as the labels equal to $0$ do not contribute to the monomial.
\begin{defi}\label{def:touch}
Let $P \coloneqq (\pi,w,dv) \in \LD(m,n)^{\bullet k}$. Define a \emph{touching point} of $P$ to be a point on the main diagonal that is the starting point of a non-decorated vertical step of $P$, labelled with a positive label. The \emph{touch} of a path $P$, denoted $\touch(P)$, is the number of touching points of $P$.
\end{defi}
For example the path in Figure~\ref{fig:labelled-dyck-path} has touch $1$.
We define two statistics on this set.
\begin{defi}\label{def:area}
For $(\pi, w, dv) \in \LD(m,n)^{\bullet k}$ we define
\[ \area(\pi, w, dv) \coloneqq \sum_{i =1}^{m+n} a_i(\pi), \]
\ie the number of whole squares between the path and the main diagonal.
\end{defi}
For example, the path in Figure~\ref{fig:labelled-dyck-path} has area $7$. Notice that the area does not depend on the labelling or decorations.
\begin{defi}\label{def:dinv}
Let $(\pi, w, dv) \in \LD(m,n)^{\bullet k}$. For $1 \leq i < j \leq n$, the pair $(i,j)$ is a \emph{diagonal inversion} if
\begin{itemize}
\item either $a_i(\pi) = a_j(\pi)$ and $w_i < w_j$ \textup{(}\emph{primary inversion}\textup{)},
\item or $a_i(\pi) = a_j(\pi) + 1$ and $w_i > w_j$ \textup{(}\emph{secondary inversion}\textup{)},
\end{itemize}
where $w_i$ denotes the $i$-th letter of $w$, \ie the label of the vertical step in the $i$-th row. Then we define
\begin{align*}
\dinv(\pi, w, dv) \coloneqq \# \{ 1 \leq i < j \leq n \mid (i,j) \text{ diagonal inversion} \land i \not \in dv \} - \# dv.
\end{align*}
\end{defi}
For example, the path in Figure has $3$ primary inversions ($(2,3),(2,5)$ and $(2,8)$), $3$ secondary inversions ($(2,4),(6,7)$ and $(6,8)$) and $2$ decorated valleys. So its dinv equals $3+3-2=4$.
It is easy to check that if $j \in dv$ then there exists some diagonal inversion $(i,j)$ and so the dinv is always non-negative (see \cite[Proposition~1]{Iraci-VandenWyngaerd-Valley-Square-2021}).
\section{The valley generalised Delta conjecture}
Now we have all the tools to state the conjectural formulas that are the object of this paper. The following conjecture was first stated in \cite{Haglund-Remmel-Wilson-2018}.
\begin{conjecture}[Delta conjecture, valley version]
\label{conj: Delta conjecture, valley version} \hfill \\
For $n,k\in \N$ with $k 0$, by Theorem~\ref{thm:sf-identity} we have
\begin{align*}
h_m^\perp & \Theta_{e_k} \nabla E_{n-k,r}\\ & = \sum_{p=0}^m t^{m-p} \sum_{i=0}^p q^{\binom{i}{2}} \qbinom{r-p+i}{i}_q \qbinom{r}{p-i}_q \Delta_{h_{m-p}} \Theta_{e_{k-i}} \nabla E_{n-m-(k-i), r-p+i}.
\end{align*}
By Theorem~\ref{thm:pushing-theorem}, we can rewrite the statement of Theorem~\ref{thm:sf-identity} as
\begin{align*}
\sum_{p=0}^m & t^{m-p} \sum_{i=0}^p q^{\binom{i}{2}} \qbinom{r-p+i}{i}_q \qbinom{r}{p-i}_q \sum_{\substack{P \in \LD(m-p, n-m )^{\bullet k-i}\\ \touch(P)= r-p+i}} q^{\dinv(P)} t^{\area(P)} x^P\\ & = \sum_{p=0}^m t^{m-p} \sum_{i=0}^p q^{\binom{i}{2}} \qbinom{r-p+i}{i}_q \qbinom{r}{p-i}_q \Delta_{h_{m-p}} \Theta_{e_{k-i}} \nabla E_{n-m-(k-i), r-p+i}.
\end{align*}
By induction hypothesis, whenever $p > 0$ we have \[ \Delta_{h_{m-p}} \Theta_{e_{k-i}} \nabla E_{n-m-(k-i), r-p+i} = \sum_{\substack{P \in \LD(m-p, n-m )^{\bullet k-i}\\ \touch(P) = r-p+i}} q^{\dinv(P)} t^{\area(P)} x^P \]
so all the terms of the sum except the one when $p=0$ cancel out, and we are left with \[ t^m \sum_{\substack{P \in \LD(m, n-m)^{\bullet k}\\ \touch(P)= r}} q^{\dinv(P)} t^{\area(P)} x^P = t^m \Delta_{h_m} \Theta_{e_k} \nabla E_{n-m-k, r} \] which is, up to the substitution $n \mapsto n+m$ and a division by $t^m$, exactly what we wanted to show.
\end{proof}
\longthanks{
The authors would like to thank Michele D'Adderio for the many interesting discussions on the topic, and the anonymous referees for their useful comments and suggestions.
}
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