\begin{definition}
	Given two sequences $(a_1, \dots, a_m)$, $(b_1, \dots, b_n)$ of pairwise distinct elements, their \emph{shuffle} $(a_1, \dots, a_m) \shuffle (b_1, \dots, b_n)$ is the set of sequences $(c_1, \dots, c_{m+n})$ such that
	\begin{itemize}
		\item $\{ c_k \mid 1 \leq k \leq m+n \} = \{ a_i, b_j \mid 1 \leq i \leq m, \; 1 \leq j \leq n \}$,
		\item $c_r = a_i, \; c_s = a_j, \; i < j \implies r < s$,
		\item $c_r = b_i, \; c_s = b_j, \; i < j \implies r < s$,
	\end{itemize}
	i.e. it is the set of sequences obtained by interlacing the elements of the two starting sequences while preserving their relative order.
\end{definition}

\begin{definition}
	Given $\mu \vdash n-d$ and $\nu \vdash d$, a $\mu,\nu$-\emph{shuffle} is a sequence of numbers from $1$ to $n$ in
	\begin{align*}
	(1, \dots, \mu_1) & \shuffle \dots \shuffle (n-\mu_{\ell(\mu)}+1, \dots, n-d) \\
	& \shuffle (n-d+\nu_1, \dots, n-d+1) \shuffle \dots \shuffle (n, \dots, n-\nu_{\ell(\nu)}+1)
	\end{align*}
	i.e. a shuffle of $\ell(\mu)$ increasing sequences of length $\mu_1, \dots, \mu_{\ell(\mu)}$, and $\ell(\nu)$ decreasing sequences of length $\nu_1, \dots, \nu_{\ell(\nu)}$, obtained by picking every time the smallest available positive integers.
%	Given $\mu \vdash n$, a $\mu$-shuffle is a string of numbers from $1$ to $n$ such that the substrings $(1, \dots, \mu_1)$, $(\mu_1+1, \dots, \mu_1+\mu_2)$, $\dots$, $(n-\mu_{\ell(\mu)}+1, \dots, n)$ appear in increasing order. Given $\mu \vdash n-d$ and $\nu \vdash d$, a $\mu,\nu$-shuffle is a string of numbers from $1$ to $n$ such that the substrings $(1, \dots, \mu_1)$,  $\dots$, $(n-d-\mu_{\ell(\mu)}+1, \dots, n-d)$ appear in increasing order, and the substrings $(n-d+\nu_1, \dots, n-d+1)$, $\dots$, $(n, \dots, n-\nu_{\ell(\nu)}+1)$ appear in decreasing order.
\end{definition}

It is well known (cf. \cite[Chapter~6]{Haglund-Book-2008}) that taking the scalar product of the combinatorial side of any of the Delta conjectures with $e_\mu \cdot h_\nu$ corresponds to taking the subsets of paths whose reading word is a $\mu,\nu$-shuffle.

The following theorem is proved in \cite{DAdderio-Iraci-VandenWyngaerd-GenDeltaSchroeder}. It extends the case $m=0$ proved in \cite{DAdderio-VandenWyngaerd-2017}.

\begin{theorem}
For $m,n,k,d\in \mathbb{N}$ with $n>k\geq 0$, $n\geq d\geq 0$ and $m\geq 0$
\begin{equation}
\<\Delta_{h_m}\Delta_{e_{n-k-1}}'e_n,e_{n-d}h_d\> = \mathop{\sum_{D\in \PLD(m,n)^{\ast k}}}_{\sigma(D)\text{ is a }(n-d),(d)\text{-shuffle}} q^{\dinv(D)} t^{\area(D)}x^D.
\end{equation}
\end{theorem}

The following theorem is proved in \cite{DAdderio-Iraci-VandenWyngaerd-GenDeltat0}. It extends the case $m=0$ first proved in \cite{Garsia-Haglund-Remmel-Yoo-2017}, giving a new independent proof of that special case (see also \cite{Haglund-Rhoades-Shimozono-arxiv} and \cite{DAdderio-Iraci-VandenWyngaerd-Delta-Square} for alternative proofs).
\begin{theorem}
	For $m,n,k,d\in \mathbb{N}$ with $n>k\geq 0$, $n\geq d\geq 0$ and $m\geq 0$, we have
	\begin{equation}
	\left.\Delta_{h_m}\Delta_{e_{n-k-1}}'e_n\right|_{t=0} =\PLD_{\underline{x},q,0}(m,n)^{\ast k}\quad \text{ and } \quad \left.\Delta_{h_m}\Delta_{e_{n-k-1}}'e_n\right|_{q=0} =\PLD_{\underline{x},0,t}(m,n)^{\ast k}.
	\end{equation}
\end{theorem}