Cyclic sieving on permutations -- an analysis of maps and statistics in the FindStat database

We perform a systematic study of permutation statistics and bijective maps on permutations using SageMath to search the FindStat combinatorial statistics database to identify apparent instances of the cyclic sieving phenomenon (CSP). Cyclic sieving occurs on a set of objects, a statistic, and a map of order $n$ when the evaluation of the statistic generating function at the $d$th power of the primitive $n$th root of unity equals the number of fixed points under the $d$th power of the map. Of the apparent instances found in our experiment, we prove 34 new instances of the CSP, and conjecture three more. Furthermore, we prove the equidistribution of some statistics and show that some maps have the same orbit structure, thus cyclic sieving holds for more even more pairs of a map and a statistic. The maps which exhibit the CSP include reverse/complement, rotation, Lehmer code rotation, toric promotion, and conjugation by the long cycle, as well as a map constructed by Corteel to swap the number of nestings and crossings, the invert Laguerre heap map, and a map of Alexandersson and Kebede designed to preserve right-to-left minima. Our results show that, contrary to common expectations, actions that exhibit homomesy are not necessarily the best candidates for the CSP, and vice versa.