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

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.