A new cyclic sieving phenomenon for Catalan objects

TL;DR

This paper proves a conjecture that a certain sequence of combinatorial objects exhibits the cyclic sieving phenomenon, linking algebraic actions with q-analog enumeration.

Contribution

It establishes the existence of a new cyclic sieving phenomenon for Catalan objects, confirming prior computational conjectures.

Findings

Confirmed the cyclic sieving phenomenon for a new class of Catalan objects.

Provided a rigorous proof of the conjectured cyclic sieving behavior.

Linked combinatorial actions with q-analog enumeration formulas.

Abstract

Based on computational experiments, Jim Propp and Vic Reiner suspected that there might exist a sequence of combinatorial objects $X_n$, each carrying a natural action of the cyclic group $C_{n-1}$ of order $n-1$ such that the triple $\left(X_n,C_{n-1},\frac{1}{[n+1]_q}{2n \brack n}_q\right)$ exhibits the cyclic sieving phenomenon. We prove their suspicion right.