Weighted-Inversion Statistics And Their Symmetry Groups

. A statistic w on Sn is a weighted-inversion (w-i) statistic if there exist weights w i;j such that w(oe) = P i!j [oe i ? oe j ]w i;j for each oe 2 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 OE : Sn ! Sn with an even stronger property: given any w-i statistic w, the statistic w ffi OE 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 ..