Rational noncrossing partitions for all coprime pairs

For coprime positive integers $a \lt b$, Armstrong, Rhoades, and Williams (2013) defined a set $NC (a, b)$ of rational noncrossing partitions, a subset of the ordinary noncrossing partitions of $\lbrace 1, \dotsc , b-1 \rbrace$. Bodnar and Rhoades (2015) confirmed their conjecture that $NC(a, b)$ is closed under rotation and proved an instance of the cyclic sieving phenomenon for this rotation action. We give a definition of $NC(a, b)$ which works for all coprime $a$ and $b$ and prove closure under rotation and cyclic sieving in this more general setting. We also generalize noncrossing parking functions to all coprime $a$ and $b$, and provide a character formula for the action of $\mathfrak{S}_a \times \mathbb{Z}_{b-1}$ on $\mathsf{Park}^{NC} (a, b)$.

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Received 6 February 2017

Published 25 January 2019