Cyclic sieving and rational Catalan theory

TL;DR

This paper proves that rational noncrossing partitions are closed under rotation and exhibit the cyclic sieving phenomenon, extending classical combinatorial structures to a rational setting and confirming a conjecture in the field.

Contribution

It establishes the rotational closure and cyclic sieving phenomenon for rational noncrossing partitions, and introduces a rational generalization of noncrossing parking functions.

Findings

Proved closure of rational noncrossing partitions under rotation.

Established cyclic sieving phenomenon for these partitions.

Defined a rational analogue of noncrossing parking functions.

Abstract

Let $a < b$ be coprime positive integers. Armstrong, Rhoades, and Williams defined a set $\mathsf{NC}(a,b)$ of `rational noncrossing partitions', which form a subset of the ordinary noncrossing partitions of $\{1, 2, \dots, b-1\}$. Confirming a conjecture of Armstrong et. al., we prove that $\mathsf{NC}(a,b)$ is closed under rotation and prove an instance of the cyclic sieving phenomenon for this rotational action. We also define a rational generalization of the $\mathfrak{S}_a$-noncrossing parking functions of Armstrong, Reiner, and Rhoades.