Product formulas for certain skew tableaux

TL;DR

This paper proves conjectured product formulas for the number of standard Young tableaux of specific skew shapes using Selberg-type integrals, generalizes MacMahon's box theorem, and extends trace generating function results.

Contribution

It provides the first proofs of conjectured product formulas for certain skew shapes and introduces new generalizations in the enumeration of Young tableaux.

Findings

Proved conjectures for specific skew shape tableaux counts

Derived a generalized MacMahon's box theorem

Established a new product formula for trace generating functions

Abstract

The hook length formula gives a product formula for the number of standard Young tableaux of a partition shape. The number of standard Young tableaux of a skew shape does not always have a product formula. However, for some special skew shapes, there is a product formula. Recently, Morales, Pak and Panova joint with Krattenthaler conjectured a product formula for the number of standard Young tableaux of shape $\lambda/\mu$ for $\lambda=((2a+c)^{c+a},(a+c)^a)$ and $\mu=(a+1,a^{a-1},1)$. They also conjectured a product formula for the number of standard Young tableaux of a certain skew shifted shape. In this paper we prove their conjectures using Selberg-type integrals. We also give a generalization of MacMahon's box theorem and a product formula for the trace generating function for a certain skew shape, which is a generalization of a recent result of Morales, Pak and Panova.