Proof of a conjecture of Morales-Pak-Panova on reverse plane partitions

TL;DR

This paper proves a conjecture relating reverse plane partitions of staircase shapes to a determinant formula involving $q$-analogues of Euler numbers, advancing understanding of combinatorial enumeration and symmetric functions.

Contribution

It provides a proof of a conjecture connecting reverse plane partitions of staircase shapes with a determinant expression involving $q$-analogues of Euler numbers.

Findings

Confirmed the determinant formula for reverse plane partitions of staircase shapes.

Established the connection between $q$-analogues of Euler numbers and reverse plane partitions.

Enhanced combinatorial enumeration techniques for skew shapes.

Abstract

Using equivariant cohomology theory, Naruse obtained a hook length formula for the number of standard Young tableaux of skew shape $\lambda/\mu$. Morales, Pak and Panova found two $q$-analogues of Naruse's formula respectively by counting semistandard Young tableaux of shape $\lambda/\mu$ and reverse plane partitions of shape $\lambda/\mu$. When $\lambda$ and $\mu$ are both staircase shape partitions, Morales, Pak and Panova conjectured that the generating function of reverse plane partitions of shape $\lambda/ \mu$ can be expressed as a determinant whose entries are related to $q$-analogues of the Euler numbers. The objective of this paper is to prove this conjecture.