Generalization of Knuth's formula for the number of skew tableaux

TL;DR

This paper generalizes Knuth's formula for counting skew tableaux by deriving explicit formulas for Kostka numbers with specific shapes and weights, using an elementary approach based on Lassalle's formula.

Contribution

It introduces a new generalization of Knuth's formula for Kostka numbers with particular shapes and weights, expanding the combinatorial understanding of skew tableaux.

Findings

Derived explicit formulas for Kostka numbers with shapes  and weights (m,1^{n-m}) for m=3,4

Provided an elementary derivation using Lassalle's explicit formula

Extended the combinatorial enumeration of skew tableaux

Abstract

We take an elementary approach to derive a generalization of Kunth's formula using Lassalle's explicit formula. In particular, we give a formula for the Kostka numbers of a shape $\mu\vdash n$ and weight $(m,1^{n-m})$ for $m=3,\;4$.