A property of dominance of partitions

TL;DR

The paper investigates a property of integer partitions related to dominance relations preserved under a specific reordering operation involving a parameter k.

Contribution

It establishes that dominance relations between partitions are maintained after applying the ^{(k)} transformation, revealing a new structural property of partitions.

Findings

Dominance is preserved under the ^{(k)} transformation.

The property holds for partitions of the same weight.

The result provides insight into the structure of partitions under reordering.

Abstract

Given an integer partition $\la=(\la_1, ..., \la_\ell)$ and an integer k, denote by $\la^{(k)}$ the sequence of length $\ell$ obtained by reordering the values $|\la_i-k|$ in non-increasing order. If $\la$ dominates $\mu$ and has the same weight, then $\la^{(k)}$ dominates $\mu^{(k)}$.