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 Xₙ), which is a class of set partitions of $[n-1]$. More precisely, Thiel proved that, with a natural action of the cyclic group Cₙ₋₁ on Xₙ, the triple (Xₙ,Cₙ₋₁,Catₙ(q)) exhibits the CSP, where Catₙ(q):=1/[n+1]qbmatrix 2n\\ n bmatrixq is MacMahon's q-Catalan number. Recently, in a study of the fermionic diagonal coinvariant ring FDRₙ, J. Kim found a combinatorial basis for FDRₙ indexed by Xₙ. In this paper, we continue to study Xₙ and obtain the following results: (1) We define a statistic $cwt$ on Xₙ whose generating function is Catₙ(q), which answers a problem of Thiel. (2) We show that Catₙ(q) is equivalent to ∑_k,x,y\\2k+x+y=n-1bmatrix n-1 2k,x,y bmatrixqCatₖ (q)q^k+x2+y2+n2 modulo qⁿ⁻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 I₂(n-1) (for even n), we prove a dihedral sieving result on Xₙ.