Polynomial Expressions for the Dimensions of the Representations of Symmetric Groups and Restricted Standard Young Tableaux

TL;DR

This paper explores polynomial formulas for the dimensions of symmetric group representations and standard Young tableaux, revealing bounds, combinatorial interpretations, and invariance properties of these polynomials.

Contribution

It introduces bounds on polynomial expansions of representation dimensions and provides combinatorial interpretations and bijections demonstrating invariance.

Findings

Polynomial expansion coefficients are bounded by the length of the partition.

Coefficients count standard Young tableaux with specific increasing row conditions.

The number of tableaux counted is independent of the choice of consecutive numbers.

Abstract

Given a partition $\lambda$ of a number $k$, it is known that by adding a long line of length $n-k$, the dimension of the associated representation of $S_{n}$ is an integer-valued polynomial of degree $k$ in $n$. We show that its expansion in the binomial basis is bounded by the length of $\lambda$, and that the resulting coefficient of index $h$, with alternating signs, counts the standard Young tableaux of shape $\lambda$ in which a given collection of consecutive $h$ numbers lie in increasing rows. We also construct bijections in order to demonstare explicitly that this number is indeed independent of the set of consecutive $h$ numbers used.