Séminaire Lotharingien de Combinatoire, 80B.54 (2018), 12 pp.
Hwang, Jang Soo
Yoo and Sun-mi
Reverse Plane Partitions of Skew Staircase Shapes and q-Euler Numbers
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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