Dihedral sieving on cluster complexes

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