zbMATH — the first resource for mathematics

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.

05A05 Permutations, words, matrices
05A16 Asymptotic enumeration
Full Text: DOI arXiv
[1] Baxter, A. and Zeilberger, D. (2011) The Number of Inversions and the Major Index of Permutations are Asymptotically Joint-Independently Normal (Second Edition!), Personal Journal of Shalosh B. Ekhad and Doron Zeilberger.
[2] Canfield, E., Janson, S. and Zeilberger, D. (2009) The Mahonian probability distribution on words is asymptotically normal. Adv. Appl. Math.46109-124. · Zbl 1227.05009
[3] Feller, W. (1957) An Introduction to Probability Theory and its Applications, Vols 1 and 2, second edition, Wiley. · Zbl 0158.34902
[4] Foata, D., On the Netto inversion number of a sequence, Proc. Amer. Math. Soc., 19, 236-240, (1968) · Zbl 0157.03403
[5] Foata, D. and Han, G.-N. (2009) The q-series in combinatorics: Permutation statistics (preliminary version). Unpublished note available at http://www-irma.u-strasbg.fr/ foata/qseries.html.
[6] Graham, R., Knuth, D. and Patashnik, O. (1989) Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley.1001562
[7] Isserlis, L., On a formula for the product-moment coefficient of any order of a normal frequency distribution in any number of variables, Biometrika, 12, 134-139, (1918)
[8] Macmahon, P., The indices of permutations and the derivation therefrom of functions of a single variable associated with the permutations of any assemblage of objects, Amer. J. Math., 35, 281-322, (1913) · JFM 44.0076.02
[9] Rahman, M. and Verma, A. (1993) Quadratic transformation formulas for basic hypergeometric series. Trans. Amer. Math. Soc.335277-302. doi:10.1090/S0002-9947-1993-1074149-81074149 · Zbl 0767.33011
[10] Slater, L. J., Generalized Hypergeometric Functions, (1966), Cambridge University Press · Zbl 0135.28101
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.