Hostname: page-component-74656f9884-2dmjx Total loading time: 0 Render date: 2026-11-11T02:59:48.532Z Has data issue: false hasContentIssue false
  • English
  • Français

The Inversion Number and the Major Index are Asymptotically Jointly Normally Distributed on Words

Part of: Combinatorics

Published online by Cambridge University Press:  14 January 2016

MARKO THIEL
MARKO THIEL*
Affiliation:
Institut für Mathematik, Universität Zürich (e-mail: markth@math.uzh.ch)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

In a recent paper, Baxter and Zeilberger 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 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 classification

Primary: 05A05: Permutations, words, matrices
Secondary: 05A16: Asymptotic enumeration

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 25 , Issue 3 , May 2016 , pp. 470 - 483
Copyright
Copyright © Cambridge University Press 2016 

References

[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.Google Scholar
[2] Canfield, E., Janson, S. and Zeilberger, D. (2009) The Mahonian probability distribution on words is asymptotically normal. Adv. Appl. Math. 46 109–124.CrossRefGoogle Scholar
[3] Feller, W. (1957) An Introduction to Probability Theory and its Applications, Vols 1 and 2, second edition, Wiley.Google Scholar
[4] Foata, D. (1968) On the Netto inversion number of a sequence. Proc. Amer. Math. Soc. 19 236–240.Google Scholar
[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 Unpublished note available at http://www-irma.u-strasbg.fr/~foata/qseries.html.Google Scholar
[6] Graham, R., Knuth, D. and Patashnik, O. (1989) Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley.Google Scholar
[7] Isserlis, L. (1918) 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.CrossRefGoogle Scholar
[8] MacMahon, P. (1913) 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.Google Scholar
[9] Rahman, M. and Verma, A. (1993) Quadratic transformation formulas for basic hypergeometric series. Trans. Amer. Math. Soc. 335 277–302.Google Scholar
[10] Slater, L. J. (1966) Generalized Hypergeometric Functions, Cambridge University Press.Google Scholar