2 3 Ju n 19 94 GRAPHICAL MAJOR INDICES

A generalization of the classical statistics “maj” and “inv” (the major index and number of inversions) on words is introduced, parameterized by arbitrary graphs on the underlying alphabet. The question of characterizing those graphs that lead to equi-distributed “inv” and “maj” is posed and answered. Résumé: On introduit une généralisation des statistiques classiques que sont “maj” et “inv” (l’indice majeur et le nombre d’inversions) sur les mots, qui est paramétrisée par des graphes arbitraires sur l’alphabet sousjacent. La question de caractériser ces graphes conduisant à des statistiques “inv” et “maj” qui soient équidistribuées est posée et résolue. 0. Introduction Every mathematician knows what the the number of inversions of a permutation is, as it features in the definition of the determinant. The number of inversions of a permutation of length n, inv π = ∑ 1≤i≤j≤n χ(π(i) > π(j)), (using the classical notation χ(A) = 1 or 0, depending on whether the statement A is true of false) is a measure of how ‘scrambled’ it is compared to the identity permutation [1, 2, . . . , n]. Netto proved (and it is nowadays easy to see, e.g., [Kn73, p. 15]) that the generating function for “the number of inversions”