Permutation Statistics in Conjugacy Classes of the Symmetric Group

We introduce the notion of a weighted inversion statistic on the symmetric group, and examine its distribution on each conjugacy class. Our work generalizes the study of several common permutation statistics, including the number of inversions, the number of descents, the major index, and the number of excedances. As a consequence, we obtain explicit formulas for the first moments of several statistics by conjugacy class. We also show that when the cycle lengths are sufficiently large, the higher moments of arbitrary permutation statistics are independent of the conjugacy class. Fulman (J. Comb. Theory Ser. A., 1998) previously established this result for major index and descents. We obtain these results, in part, by generalizing the techniques of Fulman (ibid.), and introducing the notion of permutation constraints. For permutation statistics that can be realized via symmetric constraints, we show that each moment is a polynomial in the degree of the symmetric group.
Comment: We would also like to express our gratitude to Yan Zhuang for kindly alerting us to the arXiv paper of Hamaker and Rhoades (arXiv:2206.06567), after seeing the first version of the present paper. We also thank Zach Hamaker for taking the time to explain the results of the Hamaker--Rhoades paper and its overlap with the present work