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

TL;DR

This paper connects $q$-Euler numbers to known permutation statistics, proves two conjectures on reverse plane partitions of skew staircase shapes, and derives related formulas using continued fractions and bijections.

Contribution

It proves two conjectures on RPPs of skew staircase shapes and links $q$-Euler numbers to permutation statistics through bijections and continued fraction techniques.

Findings

Proved two conjectures on RPPs of skew staircase shapes.

Connected $q$-Euler numbers to permutation statistics involving inv and maj.

Derived formulas using continued fractions and bijections.

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.