We recall that taking a scalar product 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-Delta-Square}. It extends the case $m=k=d=0$ proved by Can and Loehr in \cite{Can-Loehr-2006}.
\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}
\left\<\frac{[n-k]_t}{[n]_t}\Delta_{h_m}\Delta_{e_{n-k}}\omega(p_n),e_{n-d}h_d\right\> = \mathop{\sum_{P\in \PLSQE(m,n)^{\ast k}}}_{\sigma(P)\text{ is a }(n-d),(d)\text{-shuffle}} q^{\dinv(P)} t^{\area(P)}x^P.
\end{equation}
\end{theorem}

The following theorem is proved in \cite{DAdderio-Iraci-VandenWyngaerd-Delta-Square}.
\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.\frac{[n-k]_t}{[n]_t}\Delta_{h_m}\Delta_{e_{n-k}}\omega(p_n)\right|_{q=0} =\PLSQE_{\underline{x},0,t}.
	\end{equation}
\end{theorem}
Observe that the case $t=0$ of the generalized Delta square conjecture remains open.