A new cyclic sieving phenomenon for Catalan objects

Based on computational experiments, Jim Propp and Vic Reiner suspected that there might exist a sequence of combinatorial objects Xn, each carrying a natural action of the cyclic group Cn−1 of order n−1 such that the triple Xn,Cn−1,1[n+1]q2nnq exhibits the cyclic sieving phenomenon. We prove their suspicion right.