The Mahonian probability distribution on words is asymptotically normal.

*(English)*Zbl 1227.05009
Adv. Appl. Math. 46, No. 1-4, 109-124 (2011); corrigendum ibid. 49, No. 1, 77 (2012).

Editorial remark: According to the corrigendum, notions and results from the authors’ work are well-known (see, e.g., {[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).

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 \(\binom {a_1+\cdots +a_m}{a_1,\cdots ,a_m}\), 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.

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).

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 \(\binom {a_1+\cdots +a_m}{a_1,\cdots ,a_m}\), 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.

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).

##### MSC:

05A05 | Permutations, words, matrices |

05A16 | Asymptotic enumeration |

05E99 | Algebraic combinatorics |

68W30 | Symbolic computation and algebraic computation |

##### Keywords:

mahonian statistics; Gaussian polynomials; central and local limit theorem; symbolic computation##### References:

[1] | Andrews, G.E., The theory of partitions, (1976), Addison-Wesley Reading, MA · Zbl 0371.10001 |

[2] | Bender, E.A., Central and local limit theorems applied to asymptotic enumeration, J. combin. theory ser. A, 15, 91-111, (1973) · Zbl 0242.05006 |

[3] | Feller, W., An introduction to probability theory and its application, vol. I, (1968), Wiley New York · Zbl 0155.23101 |

[4] | Feller, W., An introduction to probability theory and its applications, vol. II, (1971), Wiley New York · Zbl 0219.60003 |

[5] | Louchard, G.; Prodinger, H., The number of inversions in permutations: a saddle point approach, J. integer seq., 6, (2003), Article 03.2.8 · Zbl 1024.05006 |

[6] | OʼHara, K.M., Unimodality of Gaussian coefficients: a constructive proof, J. combin. theory ser. A, 53, 29-52, (1990) · Zbl 0697.05002 |

[7] | Schur, I., Vorlesungen über invariantentheorie, () · Zbl 0159.03703 |

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