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The inversion number and the major index are asymptotically jointly normally distributed on words. (English) Zbl 1372.05005
Summary: In a recent paper, A. Baxter and D. Zeilberger [“The number of inversions and the major index of permutations are asymptotically joint-independently normal”, arXiv:1004.1160] showed that the two most important Mahonian statistics, the inversion number and the major index, are asymptotically independently normally distributed on permutations. In another recent paper, Canfield, Janson and Zeilberger E. R. Canfield et al. [Adv. Appl. Math. 46, No. 1–4, 109–124 (2011; Zbl 1227.05009); corrigendum ibid. 49, No. 1, 77 (2012; Zbl 1242.05005)] proved the result, already known to statisticians, that the Mahonian distribution is asymptotically normal on words. This leaves one question unanswered: What, asymptotically, is the joint distribution of the inversion number and the major index on words? We answer this question by establishing convergence to a bivariate normal distribution.

MSC:
05A05 Permutations, words, matrices
05A16 Asymptotic enumeration
Software:
MahonianStat
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References:
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