Adams et al.'s cyclic sieving conjecture for Statistic 373

A permutation statistic is a map from permutations to nonnegative integers, and two statistics are equidistributed when they have the same value-distribution on permutations. The Corteel involution is the involution on permutations described in the paper, and the cyclic sieving phenomenon concerns the fixed-point counts of powers of an involution.

Adams et al.'s conjecture. The number of weak excedances that are also mid-points of a decreasing subsequence of length $3$ (Statistic $373$ in FindStat) is equidistributed with the cycle descent number (Statistic $317$ in FindStat), and thus exhibits the cyclic sieving phenomenon under involutions with $1$ point for $n=0$ and $2^{n-1}$ points for all $n\geq 1$. In particular, this statistic exhibits the cyclic sieving phenomenon with respect to the Corteel involution.

This conjecture concerns an instance of cyclic sieving for permutation statistics and is presented as the conjecture proved in the paper. Its relationship with the Corteel involution places it in the paper's broader program of constructing cyclic sieving phenomena using continued-fraction methods.