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.
Highlights
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
This paper is about generalized noncrossing partitions arising in rational Catalan combinatorics
When m = 1, this reduces to the classical Catalan number
Summary
This paper is about generalized noncrossing partitions arising in rational Catalan combinatorics. The construction of NC(a, b) in [4] was indirect and involved the intermediate object of rational Dyck paths This left open the question of whether many of the fundamental properties of classical noncrossing partitions generalize to the rational case. It was unknown whether the set NC(a, b) is closed under the dihedral group of symmetries of the disk with b − 1 labeled boundary points. The more important of these will involve an idea genuinely new to rational Catalan combinatorics: a new measure of size for blocks of set partitions in NC(a, b) called rank.
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