Weak Hopf algebras and the distribution of involutions in symmetric groups

TL;DR

This paper establishes a connection between weak Hopf algebras and the enumeration of involutions in symmetric groups, providing a new formula for counting involutions within specific cosets related to Young subgroups.

Contribution

It introduces a novel approach using Frobenius-Schur indicators of weak Hopf algebra modules to derive formulas for involution counts in symmetric groups.

Findings

Derived a formula for involutions in symmetric groups via weak Hopf algebra modules

Connected Frobenius-Schur indicators to combinatorial enumeration of involutions

Applied the method to involutions in cosets with respect to Young subgroups

Abstract

By computing Frobenius-Schur indicators of modules of certain weak Hopf algebras, we give a formula for the number of involutions in symmetric groups, which are contained in a given coset with respect to a given Young subgroup.