Conjectured statistics for the higher \(q,t\)-Catalan sequences (Q1773200)

\textit{A. Garsia} and \textit{M. Haiman} [J. Algebr. Comb. 5, 191--244 (1996; Zbl 0853.05008)] introduced both \(q,t\)-Catalan sequences and higher \(q,t\)-Catalan sequences. \textit{J. Haglund} [Adv. Math. 175, 319--334 (2003; Zbl 1043.05012)] and Haiman have independently introduced different combinatorial statistics for \(q,t\)-Catalan sequences. The paper under review conjectures similar statistics for the higher \(q,t\)-Catalan sequences and proves a number of summation formulas and recursions for them. The author also proves several special cases of the conjectured results and suggests approaches for proving some of the conjectures. The author further summarizes recent and related developments in the area.