In \cite{Haglund-Remmel-Wilson-2015}, 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 \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 $f$. In fact in the same article \cite{Haglund-Remmel-Wilson-2015} the authors conjectured a combinatorial formula for the more general $\Delta_{h_m}\Delta_{e_{n-k-1}}'e_n$ in terms of decorated partially labelled Dyck paths, which we call \emph{generalized Delta conjecture}.

The special case $m=k=0$ gives precisely the \emph{Shuffle conjecture} in \cite{HHLRU-2005}, now a theorem of Carlsson and Mellit \cite{Carlsson-Mellit-ShuffleConj-2015}. The latter turns out to be a combinatorial formula for the Frobenius characteristic of the $\mathfrak{S}_n$-module of diagonal harmonics studied by Garsia and Haiman in relation to the famous \emph{$n!$ conjecture}, now $n!$ theorem of Haiman \cite{Haiman-nfactorial-2001}.


In \cite{Loehr-Warrington-square-2007} Loehr and Warrington conjectured a combinatorial formula for $\Delta_{e_{n}}\omega(p_n)=\nabla \omega(p_n)$ in terms of labelled square paths (ending east), called \emph{square conjecture}. The special case $\<\cdot ,e_n\>$ of this conjecture, known as \emph{$q,t$-square}, has been proved earlier by Can and Loehr in \cite{Can-Loehr-2006}. Recently the full square conjecture has been proved by Sergel in \cite{Leven-2016} after the breakthrough of Carlsson and Mellit in \cite{Carlsson-Mellit-ShuffleConj-2015}.

In \cite{DAdderio-Iraci-VandenWyngaerd-Delta-Square}, we conjectured a combinatorial formula for $\frac{[n-k]_t}{[n]_t}\Delta_{h_m}\Delta_{e_{n-k}}\omega(p_n)$ in terms of \emph{decorated partially labelled square paths} that we call \emph{generalized Delta square conjecture}. In analogy with the Delta conjecture in \cite{Haglund-Remmel-Wilson-2015}, we call simply \emph{Delta square conjecture} the special case $m=0$. Our conjecture extends the square conjecture of Loehr and Warrington \cite{Loehr-Warrington-square-2007} (now a theorem \cite{Leven-2016}), i.e. it reduces to that one for $m=k=0$. Moreover, it extends the generalized Delta conjecture in the sense that on decorated partially labelled Dyck paths gives the same combinatorial statistics. Our conjecture answers a question in \cite{Haglund-Remmel-Wilson-2015}.

In the present article we sketch the state of the art about the generalized Delta conjecture and the generalized Delta square conjecture. In this survey we highlights some results from our more recent works \cite{DAdderio-Iraci-VandenWyngaerd-Delta-Square,DAdderio-Iraci-VandenWyngaerd-GenDeltat0,DAdderio-Iraci-VandenWyngaerd-GenDeltaSchroeder}.
