Graphical Major Indices
نویسنده
چکیده
A generalization of the classical statistics \maj" and \inv" (the major index and number of inversions) on words is introduced, parame-terized 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. R esum e: On introduit une g en eralisation des statistiques classiques que sont \maj" et \inv" (l'indice majeur et le nombre d'inversions) sur les mots, qui est param etris ee par des graphes arbitraires sur l'alphabet sous-jacent. La question de caract eriser ces graphes conduisant a des statis-tiques \inv" et \maj" qui soient equidistribu ees est pos ee et r esolue. 0. Introduction Every mathematician knows what the the number of inversions of a permutation is, as it features in the deenition of the determinant. The number of inversions of a permutation of length n, inv = X 1ijn ((i) > (j)); (using the classical notation (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"
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