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 C₍-₁ on Xₙ, the triple (Xₙ, C₍-₁, Catₙ (q) ) exhibits the CSP, where Catₙ (q): =1n+1qbmatrix 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 ₊, ₗ, ₘ\\₂₊+ₗ+ₘ=₍-₁bmatrix n-1 2k, x, y bmatrixqCatₖ (q) q^k+x{2+y2+n2} 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 I₂ (n-1) (for even n), we prove a dihedral sieving result on Xₙ.