Reverse plane partitions of skew staircase shapes and $q$-Euler numbers

Abstract

Recently, Naruse discovered a hook length formula for the number of standard Young tableaux of a skew shape. Morales, Pak and Panova found two $q$-analogs of Naruse's hook length formula over semistandard Young tableaux (SSYTs) and reverse plane partitions (RPPs). As an application of their formula, they expressed certain $q$-Euler numbers, which are generating functions for SSYTs and RPPs of a zigzag border strip, in terms of weighted Dyck paths. They found a determinantal formula for the generating function for SSYTs of a skew staircase shape and proposed two conjectures related to RPPs of the same shape. One conjecture is a determinantal formula for the number of \emph{pleasant diagrams} in terms of Schr\"oder paths and the other conjecture is a determinantal formula for the generating function for RPPs of a skew staircase shape in terms of $q$-Euler numbers. In this paper, we show that the results of Morales, Pak and Panova on the $q$-Euler numbers can be derived from previously known results due to Prodinger by manipulating continued fractions. These $q$-Euler numbers are naturally expressed as generating functions for alternating permutations with certain statistics involving \emph{maj}. It has been proved by Huber and Yee that these $q$-Euler numbers are generating functions for alternating permutations with certain statistics involving \emph{inv}. By modifying Foata's bijection we construct a bijection on alternating permutations which sends the statistics involving \emph{maj} to the statistic involving \emph{inv}. We also prove the aforementioned two conjectures of Morales, Pak and Panova.