Reverse Plane Partitions of Skew Staircase Shapes and q-Euler Numbers

Byung-Hak Hwang   ; Jang Soo Kim   ; Meesue Yoo   ; Sun-mi Yun

Séminaire lotharingien de combinatoire, 80B (2018)

Résumé

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.

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 maj. It has been proved by Huber and Yee that these q-Euler numbers are generating functions for alternating permutations with certain statistics involving inv. By modifying Foata's bijection we construct a bijection on alternating permutations which sends the statistics involving maj to the statistic involving inv. We also prove the aforementioned two conjectures of Morales, Pak and Panova.