A proof of the \(q,t\)-square conjecture
The authors prove a combinatorial formula conjectured by \textit{N. A. Loehr} and \textit{G. S. Warrington} [Trans. Am. Math. Soc. 359, No. 2, 649--669 (2007; Zbl 1107.05098)] for the coefficient of the sign charater in \(\nabla(p_n)\), where \(\nabla\) is the Bergeron-Garsia nabla operator [\textit{F. Bergeron} and \textit{A. M. Garsia}, Algebraic methods and \(q\)-special functions. CRM Proc. Lect. Notes 22, 1--52 (1999; Zbl 0947.20009)] and \(p_n\) the power sum symmetric function. The formula is analogous to the \((q,t)\)-Catalan theorem, which givs a combinatorial formula for \(\nabla(e_n)\), where \(e_n\) is the elementary symmetric function. The combinatorial formula in the \((q,t)\)-Catalan theorem enumerates Dyck paths (lattice paths in the square which stay on or above the main diagonal) weighted by certain statistics. The analogous formula for the \((q,t)\)-square conjecture enumerates all lattice paths in the square weighted by similar statistics.
scientific article; zbMATH DE number 5078223
scientific article