A note on Kostka numbers

TL;DR

This paper provides an elementary proof of a known inequality involving Kostka numbers, showing that for partitions with a dominance relation, the Kostka number decreases accordingly.

Contribution

It offers a new, elementary proof of a classical result on Kostka numbers, addressing a question from MathOverflow.

Findings

Proves that if nd  are partitions with  ; , then K_{\u001,} ; K_{,}

Establishes the inequality K_{\u001,} ; K_{,} under dominance order

Provides insight into the structure of Kostka numbers and their monotonicity properties.

Abstract

We give an elementary proof of a well-known result on Kostka numbers, following a question from Mark Wildon on MathOverflow. Namely, we show that given partitions $\lambda,\mu,\nu$ of $n$ with $\mu\trianglerighteq\nu$, we have $K_{\lambda\nu}\geqslant K_{\lambda\mu}$.