Cyclic sieving phenomena via combinatorics of continued fractions

TL;DR

This paper explores cyclic sieving phenomena across various combinatorial objects using continued fraction identities, introducing new involutions and reinterpreting existing results to deepen understanding of symmetry and enumeration.

Contribution

It provides a unified framework for cyclic sieving involving multiple combinatorial families through continued fractions, introduces a new involution for D-permutations, and reestablishes several known results.

Findings

Reproved several cyclic sieving results for permutations using continued fractions.

Proved two conjectures related to cyclic sieving phenomena.

Constructed a new involution for D-permutations, called the Genocchi-Corteel involution.

Abstract

We will exhibit several instances of the cyclic sieving phenomenon involving statistics and involutions on the following combinatorial families of objects: permutations, set partitions, perfect matchings, D-permutations (and its subclasses). Our results will be based on continued fraction identities enumerating these objects. Our instances of cyclic sieving phenomenon for permutations involve the Corteel involution; this was first studied by Adams, Elder, Lafreni\`ere, McNicholas, Striker and Welch (arxiv~2024). We will reprove several of their results using our setting of continued fractions; we will also prove two of their conjectures. Our study of set partitions and perfect matchings will involve the Kasraoui-Zeng involution and the Chen-Deng-Du-Stanley-Yan (CDDSY) involution. Finally, for D-permutations, we will construct a new involution which we call the Genocchi-Corteel involution. The common feature of all of these involutions, other than the CDDSY involution, is that they are constructed via bijections to weighted lattice paths, and that they exchange crossings and nestings on their respective objects.