Dihedral Sieving Phenomena

TL;DR

This paper extends the cyclic sieving phenomenon to dihedral groups, exploring its connections with Fibonomial coefficients and $(q,t)$-Catalan numbers, and provides new instances of dihedral sieving.

Contribution

It introduces a definition of sieving for arbitrary groups, specifically studies dihedral sieving, and connects it with recent combinatorial polynomials.

Findings

Identifies dihedral sieving instances with Fibonomial coefficients

Demonstrates dihedral sieving involving $(q,t)$-Catalan numbers

Provides a framework for understanding sieving phenomena in dihedral groups

Abstract

Cyclic sieving is a well-known phenomenon where certain interesting polynomials, especially $q$-analogues, have useful interpretations related to actions and representations of the cyclic group. We propose a definition of sieving for an arbitrary group $G$ and study it for the dihedral group $I_2(n)$ of order $2n$. This requires understanding the generators of the representation ring of the dihedral group. For $n$ odd, we exhibit several instances of dihedral sieving which involve the generalized Fibonomial coefficients, recently studied by Amdeberhan, Chen, Moll, and Sagan. We also exhibit an instance of dihedral sieving involving Garsia and Haiman's $(q,t)$-Catalan numbers.