Counting symmetry classes of dissections of a convex regular polygon

TL;DR

This paper derives explicit formulas for counting symmetry classes of polygon dissections under cyclic and dihedral group actions, using combinatorial and group-theoretic methods, and recovers known results while providing new enumerations.

Contribution

It introduces explicit formulas for symmetry class counts of polygon dissections, employing the Cauchy-Frobenius Lemma and bijections, advancing combinatorial enumeration under symmetry groups.

Findings

Derived explicit formulas for symmetry class counts

Recovered known enumeration results

Provided several new enumeration formulas

Abstract

This paper proves explicit formulas for the number of dissections of a convex regular polygon modulo the action of the cyclic and dihedral groups. The formulas are obtained by making use of the Cauchy-Frobenius Lemma as well as bijections between rotationally symmetric dissections and simpler classes of dissections. A number of special cases of these formulas are studied. Consequently, some known enumerations are recovered and several new ones are provided.