m-Axial and m-Circular 3m-Polygons

TL;DR

This paper enumerates specific symmetric and rotationally invariant polygons with 3m vertices, providing complete classifications and formulas for their counts, advancing combinatorial understanding of symmetric polygon structures.

Contribution

It introduces new enumeration methods for polygons with particular symmetry and rotational properties, including complete classifications and explicit formulas.

Findings

Counted polygons with m symmetry axes among 3m-vertices polygons.

Enumerated polygons invariant under rotations but lacking symmetry axes.

Provided formulas for the number of such polygons.

Abstract

The present article includes the enumeration of $n$-polygons with two certain symmetry properties: For a number $3m$ of vertices, we count the $3m$-polygons with $m$ symmetry axes and the $3m$-polygons, that match after three elementary rotations, but have no symmetry axes. For those polygons we give complete lists of representatives of their equivalence-classes and closed formulas for their number.