# The Mahonian probability distribution on words is asymptotically normal

E. Rodney Canfield, Svante Janson, Doron Zeilberger

Research output: Contribution to journalArticle

6 Citations (Scopus)

### Abstract

The Mahonian statistic is the number of inversions in a permutation of a multiset with ai elements of type i, 1≤i≤m. The counting function for this statistic is the q analog of the multinomial coefficient (a1+...+am/a1,...,am), 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. 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).

Original language English (US) 109-124 16 Advances in Applied Mathematics 46 1-4 https://doi.org/10.1016/j.aam.2009.10.001 Published - Jan 1 2011

### Fingerprint

Local Limit Theorem
Probability distributions
Statistic
Probability Distribution
Multinomial Coefficients
Log-concave
Probability generating function
Counting Function
Multiset
Q-analogue
Maple
Method of Moments
Coefficient
Characteristic Function
Statistics
Central limit theorem
Normalization
Inversion
Permutation
Method of moments

### All Science Journal Classification (ASJC) codes

• Applied Mathematics

### Keywords

• Central and local limit theorem
• Gaussian polynomials
• Mahonian statistics
• Symbolic computation

### Cite this

Canfield, E. Rodney ; Janson, Svante ; Zeilberger, Doron. / The Mahonian probability distribution on words is asymptotically normal. In: Advances in Applied Mathematics. 2011 ; Vol. 46, No. 1-4. pp. 109-124.
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The Mahonian probability distribution on words is asymptotically normal. / Canfield, E. Rodney; Janson, Svante; Zeilberger, Doron.

In: Advances in Applied Mathematics, Vol. 46, No. 1-4, 01.01.2011, p. 109-124.

Research output: Contribution to journalArticle

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T1 - The Mahonian probability distribution on words is asymptotically normal

AU - Canfield, E. Rodney

AU - Janson, Svante

AU - Zeilberger, Doron

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N2 - The Mahonian statistic is the number of inversions in a permutation of a multiset with ai elements of type i, 1≤i≤m. The counting function for this statistic is the q analog of the multinomial coefficient (a1+...+am/a1,...,am), 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. 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).

AB - The Mahonian statistic is the number of inversions in a permutation of a multiset with ai elements of type i, 1≤i≤m. The counting function for this statistic is the q analog of the multinomial coefficient (a1+...+am/a1,...,am), 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. 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).

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