# Rational Noncrossing Partitions for all Coprime Pairs

## TL;DR

This paper generalizes the concept of rational noncrossing partitions and parking functions to all coprime pairs, proving their closure under rotation, cyclic sieving, and providing character formulas for related group actions.

## Contribution

It introduces a unified definition of rational noncrossing partitions for all coprime pairs and proves their properties, extending previous results limited to specific cases.

## Key findings

- Proved closure under rotation for generalized rational noncrossing partitions.
- Established cyclic sieving phenomenon in the generalized setting.
- Derived character formulas for group actions on generalized parking functions.

## Abstract

For coprime positive integers $a<b$, Armstrong, Rhoades, and Williams (2013) defined a set $NC(a,b)$ of rational noncrossing partitions, a subset of the ordinary noncrossing partitions of $\{1, \ldots, b-1\}$. Bodnar and Rhoades (2015) confirmed their conjecture that $NC(a,b)$ is closed under rotation and proved an instance of the cyclic sieving phenomenon for this rotation action. We give a definition of $NC(a,b)$ which works for all coprime $a$ and $b$ and prove closure under rotation and cyclic sieving in this more general setting. We also generalize noncrossing parking functions to all coprime $a$ and $b$, and provide a character formula for the action of $\mathfrak{S}_a \times \mathbb{Z}_{b-1}$ on $\mathsf{Park}^{NC}(a,b)$.