Product formulas for certain skew tableaux

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.

20 pages, 8 figures