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arXiv · 2011.11885

Dihedral Sieving on Cluster Complexes

Zachary Stier·Julian Wellman·Zixuan Xu
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Abstract

The cyclic sieving phenomenon of Reiner, Stanton, and White characterizes the stabilizers of cyclic group actions on finite sets using q-analogue polynomials. Eu and Fu demonstrated a cyclic sieving phenomenon on generalized cluster complexes of every type using the q-Catalan numbers. In this paper, we exhibit the dihedral sieving phenomenon, introduced for odd n by Rao and Suk, on clusters of every type. In the type A case, we show that the Raney numbers count both reflection-symmetric k-angulations of an n-gon and a particular evaluation of the (q,t)-Fuss--Catalan numbers. We also introduce a sieving phenomenon for the symmetric group, and discuss possibilities for dihedral sieving for even n.

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Zachary Stier, Julian Wellman, Zixuan Xu. 2020-11-24. Dihedral Sieving on Cluster Complexes. https://arxiv.org/abs/2011.11885

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