Minimal set of class-sums characterizing the ordinary irreducible representations of the symmetric group, and the Tarry-Escott problem

Abstract

The shape of a Young diagram Y (|Y| = n) can be specified in terms of the set of symmetric power sums over its contents, σ_l = Σ_(ij)ϵY(j − i)^l; l = 1, 2, …, n. It is remarkable that the set of power sums σ₁, σ₂, …, σ_k is sufficient to characterize the Young diagrams possessing up to n(k) boxes, where n(k) is considerably larger than k. Numerical evidence for k⩽5 is roughly consistent with $\text{n(k) ∼ 4(}\text{4}\text{3}\text{)}{}^{\mathrm{2k}}$. The lower bound n(k) > k + max(2, √k) has been derived by examination of some properties of Young diagrams with a large number of rows, and the upper bound n(k) < 2^2k+1 has been established using a variant of the Tarry-Escott problem.