Statistics for $S_n$ acting on $k$-sets

TL;DR

This paper analyzes the action of the symmetric group on k-subsets, providing explicit calculations for minimal bases, height, and irredundant bases, which are fundamental in understanding the group's permutation properties.

Contribution

It offers new precise calculations for the maximum size of minimal bases, height, and irredundant bases in the natural action of $S_n$ on k-subsets, extending previous combinatorial group theory results.

Findings

Calculated maximum size of minimal bases

Determined the height of the action

Established the maximum length of irredundant bases

Abstract

We study the natural action of $S_n$ on the set of $k$-subsets of the set $\{1,\dots, n\}$ when $1\leq k \leq \frac{n}{2}$. For this action we calculate the maximum size of a minimal base, the height and the maximum length of an irredundant base. Here a "base" is a set with trivial pointwise stabilizer, "height" is the maximum size of a subset with the property that its pointwise stabilizer is not equal to the pointwise stabilizer of any proper subset, and an "irredundant base" can be thought of as a chain of (pointwise) set-stabilizers for which all containments are proper.