Cyclic sieving on noncrossing (1,2)-configurations

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 $\text{Cat}_n(q):=\frac{1}{[n+1]_q}\begin{bmatrix} 2n\\ n \end{bmatrix}_q$ is MacMahon's $q$-Catalan number. Recently, in a study of the fermionic diagonal coinvariant ring $FDR_n$, J. Kim found a combinatorial basis for $FDR_n$ 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 $$\sum_{\substack{k,x,y\\2k+x+y=n-1}}\begin{bmatrix} n-1 2k,x,y \end{bmatrix}_q\text{Cat}_k (q)q^{k+\binom{x}{2}+\binom{y}{2}+\binom{n}{2}}$$ 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 $X_n$.