## MahonianStat

 swMATH ID: 6180 Software Authors: Canfield, E.Rodney; Janson, Svante; Zeilberger, Doron Description: The Mahonian probability distribution on words is asymptotically normal According to the corrigendum, notions and results from the authors’ work are well-known (see, e.g., {[it P. Diaconis}, Group representations in probability and statistics, IMS Lecture Notes-Monograph Series, 11. Hayward, CA: Institute of Mathematical Statistics. vi, 198 p. (1998; Zbl 0695.60012)], p. 128-129).par Summary: The Mahonian statistic is the number of inversions in a permutation of a multiset with $$a_i$$ elements of type $$i, 1 leqslant i leqslant m$$. The counting function for this statistic is the $$q$$ analog of the multinomial coefficient $$inom {a_1+cdots +a_m}{a_1,cdots ,a_m}$$, and the probability generating function is the normalization of the latter. We give two proofs that the distribution is asymptotically normal. The first is computer-assisted, based on the method of moments.par The Maple package MahonianStat, available from the webpage of this article, can be used by the reader to perform experiments and calculations. Our second proof uses characteristic functions. We then take up the study of a local limit theorem to accompany our central limit theorem. Here our result is less general, and we must be content with a conjecture about further work. Our local limit theorem permits us to conclude that the coefficients of the $$q$$-multinomial are log-concave, provided one stays near the center (where the largest coefficients reside). Homepage: http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/mahon.html Dependencies: Maple Keywords: mahonian statistics; Gaussian polynomials; central and local limit theorem; symbolic computation Related Software: FindStat; SageMath; Quicksort; F12345; P123456; P12345; P1234; F1234; F123; P123; SMCper; Maple; OEIS Cited in: 12 Publications
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### Cited by 16 Authors

 4 Janson, Svante 3 Zeilberger, Doron 2 Billey, Sara C. 2 Canfield, E. Rodney 2 Konvalinka, Matjaž 2 Kousidis, Stavros 2 Swanson, Joshua P. 1 Bliem, Thomas 1 Féray, Valentin 1 Hwang, Hsien-Kuei 1 Kahle, Thomas 1 Nakamura, Brian 1 Schulte-Geers, Ernst 1 Stump, Christian 1 Thiel, Marko 1 Zacharovas, Vytas
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### Cited in 9 Serials

 3 Advances in Applied Mathematics 2 The Electronic Journal of Combinatorics 1 Mathematics of Computation 1 Random Structures & Algorithms 1 The Annals of Applied Probability 1 Journal of Algebraic Combinatorics 1 Combinatorics, Probability and Computing 1 Séminaire Lotharingien de Combinatoire 1 Journal of Combinatorics
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### Cited in 8 Fields

 12 Combinatorics (05-XX) 6 Probability theory and stochastic processes (60-XX) 2 Order, lattices, ordered algebraic structures (06-XX) 2 Group theory and generalizations (20-XX) 2 Computer science (68-XX) 1 Number theory (11-XX) 1 Linear and multilinear algebra; matrix theory (15-XX) 1 Special functions (33-XX)