Cyclic sieving and dihedral 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 $(X_n,C_{n-1},{\text{Cat}}_n(q\left)\right)$ exhibits the CSP, where ${\text{Cat}}_n\left(q\right)≔\frac{1}{{[n+1]}_q}{\left[\begin{array}{c}2n\\ n\end{array}\right]}_q$ is MacMahon's q-Catalan number. Recently, in a study of the fermionic diagonal coinvariant ring $FD{R}_n$, Jesse 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 on $X_n$ whose generating function is ${\text{Cat}}_n\left(q\right)$, which answers a problem of Thiel.

• (2)

  We show that ${\text{Cat}}_n\left(q\right)$ is equivalent to$\sum _{\begin{array}{c}k,x,y\\ 2k+x+y=n-1\end{array}}{\left[\begin{array}{c}n-1\\ 2k,x,y\end{array}\right]}_q{\text{Cat}}_k\left(q\right)q^{k+\left(\begin{array}{c}x\\ 2\end{array}\right)+\left(\begin{array}{c}y\\ 2\end{array}\right)+\left(\begin{array}{c}n\\ 2\end{array}\right)}$ modulo $q^{n-1}-1$, which answers a problem of Jesse Kim. As mentioned by Jesse 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 $X_n$.