Cyclic sieving on noncrossing (1,2)-configurations

Abstract

Verifying a suspicion of Propp and Reiner concerning the cyclic sieving phenomenon (CSP), M. Thiel introduced a Catalan object called noncrossing $(1,2)$-configurations (denoted by $X_n$), which is a class of set partitions of $[n-1]$. More precisely, Thiel proved that, with a natural action of the cyclic group $C_{n-1}$ on $X_n$, the triple $\left(X_n,C_{n-1},{\text{Cat}}_n(q)\right)$ exhibits the CSP, where ${\text{Cat}}_n(q):=\frac{1}{[n+1{]}_q}{\left[\begin{array}{c}2n\\ n\end{array}\right]}_q$ is MacMahon's $q$-Catalan number. Recently, in a study of the fermionic diagonal coinvariant ring $FD{R}_n$, J. Kim found a combinatorial basis for $FD{R}_n$ indexed by $X_n$. In this paper, we continue to study $X_n$ and obtain the following results: (1) We define a statistic $cwt$ on $X_n$ whose generating function is ${\text{Cat}}_n(q)$, which answers a problem of Thiel. (2) We show that ${\text{Cat}}_n(q)$ is equivalent to $$\sum _{\begin{array}{c}k,x,y\\ 2k+x+y=n-1\end{array}}{\left[\begin{array}{c}n-12k,x,y\end{array}\right]}_q{\text{Cat}}_k(q)q^{k+\left(\genfrac{}{}{0ex}{}{x}{2}\right)+\left(\genfrac{}{}{0ex}{}{y}{2}\right)+\left(\genfrac{}{}{0ex}{}{n}{2}\right)}$$ modulo $q^{n-1}-1$, which answers a problem of Kim. As mentioned by Kim, this result leads to a representation theoretic proof of the above cyclic sieving result of Thiel. (3) We consider the dihedral sieving, a generalization of the CSP, which was recently introduced by Rao and Suk. Under a natural action of the dihedral group $I_2(n-1)$ (for even $n$), we prove a dihedral sieving result on $X_n$.