On the base size of the symmetric and the alternating group acting on partitions

TL;DR

This paper determines the minimal number of points needed to uniquely identify elements of symmetric and alternating groups acting on partitions of a set into equal parts, advancing understanding of their permutation actions.

Contribution

It provides explicit calculations of base sizes for symmetric and alternating groups acting on partitions, a novel contribution to permutation group theory.

Findings

Calculated base sizes for symmetric groups on partitions

Calculated base sizes for alternating groups on partitions

Enhanced understanding of permutation group actions on set partitions

Abstract

Given three positive integers $n,a,b$ with $n=ab$, we determine the base size of the symmetric group and of the alternating group of degree $n$ in their action on the set of partitions into $b$ parts having cardinality $a$.