2024/02/08 by Zeng, Chuyi, Zhang, Shiwen
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2402.05771
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]q)\beginbmatrix
2n n
\endbmatrixq 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 ∑_\substackk,x,y
2k+x+y=n-1\beginbmatrix
n-1
2k,x,y
\endbmatrixqCatk
(q)q^k+\binomx2+\binomy2+\binomn2
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.