Permutation Statistics in Conjugacy Classes of the Symmetric Group

TL;DR

This paper introduces weighted inversion statistics on the symmetric group, generalizing classical permutation statistics, and derives explicit formulas for their moments across conjugacy classes, revealing independence of higher moments from conjugacy class size.

Contribution

It generalizes permutation statistics using weighted inversion concepts and provides explicit moment formulas, extending prior results to a broader class of statistics.

Findings

Explicit formulas for first moments per conjugacy class

Higher moments become class-independent for large cycle lengths

Polynomial behavior of moments for symmetric constraint statistics

Abstract

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.