Cyclic Sieving and Rational Catalan Theory

Abstract

Let $a < b$ be coprime positive integers. Armstrong, Rhoades, and Williams (2013) defined a set 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 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.