Weighted-Inversion Statistics and Their Symmetry Groups

A statistic w on Sn is a weighted-inversion (w-i) statistic if there exist weights wi, j such that w(σ)=∑i<j χ[σi>σj] wi, j for each σ∈Sn. Two well-known examples are the major index and inversion count statistics. These two statistics share the same distribution over Sn, and many bijections Sn→Sn have been described to prove this. These bijections thus have the property that they map a certain w-i statistic to another. This paper presents the results of our search for bijections φ: Sn→Sn with an even stronger property: given any w-i statistic w, the statistic w∘φ is also a w-i statistic. Such a set of bijections forms a group, which we call the core group of Sn. We exhibit a subgroup of the core group of Sn which is isomorphic to the dihedral group Dn+1. We extend these ideas to other sets of objects, including subsets of Sn and sets of permutations of a multiset. As examples, we develop a family of subsets of Sn which has a core group isomorphic to a Weyl group of order 2n·n!, and we show that the set of permutations of the multiset {0k, 1n−k} has a core group containing Sk×Sn−k as a subgroup. We demonstrate that the core group of a set A is the group of permutations of the rows of a certain matrix H (depending only on the inversion patterns of the objects in A) which preserve the column space of H. This allows us to compute the core group with no knowledge of the actual w-i statistics involved.