The sieving phenomenon for finite groups

TL;DR

This paper extends the cyclic sieving phenomenon from cyclic groups to finite abelian and dihedral groups, proposing methods for further generalizations to nonabelian groups in combinatorics.

Contribution

It introduces a natural extension of the cyclic sieving phenomenon to finite abelian groups and explores similar results for dihedral groups, suggesting broader generalizations.

Findings

Extended sieving phenomenon to finite abelian groups

Established results for dihedral groups

Proposed approaches for nonabelian groups

Abstract

The cyclic sieving phenomenon is a well-studied occurrence in combinatorics appearing when a cyclic group acts on a finite set. In this paper, we demonstrate a natural extension of this theory to finite abelian groups. We also present a similar result for dihedral groups and suggest approaches for natural generalizations to nonabelian groups.