Séminaire Lotharingien de Combinatoire, 80B.54 (2018), 12 pp.

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

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.

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.

Received: November 14, 2017. Accepted: February 17, 2018. Final version: April 1, 2018.

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