Cyclic Sieving and Rational Catalan Theory

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, . . . , 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 Sa-noncrossing parking functions of Armstrong, Reiner, and Rhoades (2015).