Enumeration of strong dichotomy patterns

TL;DR

This paper uses advanced group action counting techniques to enumerate specific bicolor patterns with trivial symmetry, proposing a potential cyclic sieving phenomenon in combinatorics.

Contribution

It applies White's Pólya-Redfield theory to enumerate strong dichotomy patterns, introducing new counts and conjectures in combinatorial symmetry analysis.

Findings

Counted bicolor patterns with trivial automorphism groups

Proposed a conjectural cyclic sieving phenomenon

Extended enumeration methods for combinatorial patterns

Abstract

We apply the version of P\'{o}lya-Redfield theory obtained by White to count patterns with a given automorphism group to the enumeration of strong dichotomy patterns, that is, we count bicolor patterns of $\mathbb{Z}_{2k}$ with respect to the action of $\Aff(\mathbb{Z}_{2k})$ and with trivial isotropy group. As a byproduct, a conjectural instance of phenomenon similar to cyclic sieving for special cases of these combinatorial objects is proposed.