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arXiv · 1510.08502

Cyclic sieving and rational Catalan theory

Michelle Bodnar·Brendon Rhoades
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Abstract

Let $a < b$ be coprime positive integers. Armstrong, Rhoades, and Williams defined a set $\mathsf{NC}(a,b)$ 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 $\mathsf{NC}(a,b)$ 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.

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Michelle Bodnar, Brendon Rhoades. 2015-10-28. Cyclic sieving and rational Catalan theory. https://arxiv.org/abs/1510.08502

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