Refined Catalan and Narayana cyclic sieving

TL;DR

This paper advances the understanding of the cyclic sieving phenomenon (CSP) by proving new instances on Catalan objects and refining existing cases, focusing on detailed combinatorial structures like triangulations and non-crossing matchings.

Contribution

It introduces new CSP instances on Catalan objects of types A and B and refines previous results by considering additional parameters such as ears, short edges, loops, and edges.

Findings

Proved several new CSP instances on Catalan objects.

Refined existing CSP results with additional combinatorial parameters.

Enhanced understanding of CSP in the context of Catalan combinatorics.

Abstract

We prove several new instances of the cyclic sieving phenomenon (CSP) on Catalan objects of type A and type B. Moreover, we refine many of the known instances of the CSP on Catalan objects. For example, we consider triangulations refined by the number of "ears", non-crossing matchings with a fixed number of short edges, and non-crossing configurations with a fixed number of loops and edges.