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 λ/µ. Morales, Pak and Panova found two q-analogues of Naruse's formula respectively by counting semistandard Young tableaux of shape λ/µ and reverse plane partitions of shape λ/µ. When λ and µ are both staircase shape partitions, Morales, Pak and Panova conjectured that the generating function of reverse plane partitions of shape λ/µ 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.

Negative moments of orthogonal polynomials

“…In this section we show that the general reciprocity theorem (Theorem 6.1) implies the following result of Cigler and Krattenthaler [2,Theorem 34]. Using this theorem we give a generalization of a result on reverse plane partitions, which was conjectured by Morales, Pak, and Panova [12] and proved independently by Hwang et al [7] and Guo et al [6].…”

“…In Section 8 we show that Theorem 6.1 also implies the weighted version of Theorem 1.3 due to Cigler and Krattenthaler [2,Theorem 34]. We then show in Theorem 8.5 that this weighted version gives a bounded and multivariate generalization of the Morales-Pak-Panova ex-conjecture [12] on reverse plane partitions, which has been proved by Hwang et al [7] and Guo et al [6] independently.…”

“…Morales, Pak, and Panova [12] conjectured the following identity, which was proved independently by Hwang et al [7] and Guo et al [6]: π∈RPP(δn+2m/δn) q |π| = q − m(m+1)(6n+8m−5) where δ n = (n − 1, . .…”

Negative moments of orthogonal polynomials

“…In this section, we show that the general reciprocity theorem (Theorem 6.1) implies the following result of Cigler and Krattenthaler [3,Theorem 34]. Using this theorem, we give a generalization of a result on reverse plane partitions, which was conjectured by Morales, Pak and Panova [14] and proved independently by Hwang et al [9] and Guo et al [7]. Proof.…”

“…In Section 8, we show that Theorem 6.1 also implies the weighted version of Theorem 1.3 due to Cigler and Krattenthaler [3,Theorem 34]. We then show in Theorem 8.5 that this weighted version gives a bounded and multivariate generalization of the Morales-Pak-Panova ex-conjecture [14] on reverse plane partitions, which has been proved by Hwang et al [9] and Guo et al [7], independently.…”