Cyclic sieving and rational Catalan theory (Q281582)

Summary: Let \(a < b\) be coprime positive integers. \textit{D. Armstrong} et al. [Electron. J. Comb. 20, No. 3, Research Paper P54, 27 p. (2013)] defined a set \(\operatorname{NC}(a,b)\) of `rational noncrossing partitions', which form a subset of the ordinary noncrossing partitions of \(\{1, 2, \ldots, b-1\}\). Confirming a conjecture of D. Armstrong et. al. [loc. cit.], we prove that \(\operatorname{NC}(a,b)\) 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 \textit{D. Armstrong} et al. [Adv. Math. 269, 647--706 (2015; Zbl 1347.20039)].