### 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) |
---|---|

Pages (from-to) | 109-124 |

Number of pages | 16 |

Journal | Advances in Applied Mathematics |

Volume | 46 |

Issue number | 1-4 |

DOIs | |

State | Published - Jan 1 2011 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Applied Mathematics

### Keywords

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

### Cite this

*Advances in Applied Mathematics*,

*46*(1-4), 109-124. https://doi.org/10.1016/j.aam.2009.10.001

}

*Advances in Applied Mathematics*, vol. 46, no. 1-4, pp. 109-124. https://doi.org/10.1016/j.aam.2009.10.001

**The Mahonian probability distribution on words is asymptotically normal.** / Canfield, E. Rodney; Janson, Svante; Zeilberger, Doron.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The Mahonian probability distribution on words is asymptotically normal

AU - Canfield, E. Rodney

AU - Janson, Svante

AU - Zeilberger, Doron

PY - 2011/1/1

Y1 - 2011/1/1

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).

KW - Central and local limit theorem

KW - Gaussian polynomials

KW - Mahonian statistics

KW - Symbolic computation

UR - http://www.scopus.com/inward/record.url?scp=79953699331&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79953699331&partnerID=8YFLogxK

U2 - 10.1016/j.aam.2009.10.001

DO - 10.1016/j.aam.2009.10.001

M3 - Article

AN - SCOPUS:79953699331

VL - 46

SP - 109

EP - 124

JO - Advances in Applied Mathematics

JF - Advances in Applied Mathematics

SN - 0196-8858

IS - 1-4

ER -