The cyclic sieving phenomenon on circular Dyck paths

TL;DR

This paper introduces a $q$-enumeration for circular Dyck paths, extends it to M"obius paths, and demonstrates the cyclic sieving phenomenon, also exploring generalizations like subset cyclic sieving and Lyndon-like cyclic sieving.

Contribution

It provides a new $q$-enumeration for circular Dyck paths and generalizes cyclic sieving to M"obius paths and other variants, expanding understanding of cyclic sieving phenomena.

Findings

$q$-enumeration of circular Dyck paths established

Cyclic sieving phenomenon demonstrated under natural cyclic group action

Generalizations to M"obius paths and subset cyclic sieving discussed

Abstract

We give a $q$-enumeration of circular Dyck paths, which is a superset of the classical Dyck paths enumerated by the Catalan numbers. These objects have recently been studied by Alexandersson and Panova. Furthermore, we show that this $q$-analogue exhibits the cyclic sieving phenomenon under a natural action of the cyclic group. The enumeration and cyclic sieving is generalized to M\"obius paths. We also discuss properties of a generalization of cyclic sieving, which we call subset cyclic sieving. Finally, we also introduce the notion of Lyndon-like cyclic sieving that concerns special recursive properties of combinatorial objects exhibiting the cyclic sieving phenomenon.