The Mahonian Probability Distribution on Words is Asymptotically Normal

By E. Rodney Canfield, Svante Janson, and Doron Zeilberger

[Appeared in Advances in Applied Mathematics v. 46 (2011), 109-124 (The Dennis Stanton special issue)]

.pdf   .ps   .tex  
First Written: Aug. 14, 2009. Revised Version: Sept. 22, 2009.
Last Update of this webpage: Feb. 10, 2012 (putting a link to an erratum)
Dedicated to Dennis Stanton, q-grandmaster and versatile unimodaler (and log-concaviter).
What is the exact number of words with zillion 1's, zillion 2's, zillion 3's, and zillion 4's, that have exactly three zillion2 inversions? No one knows exactly, but the present work, that proves that the so-called Mahonian distribution is asymptotically normal, can give you a very good approximation.

Important: This article is accompanied by Maple package MahonianStat that tells everything you'd like to know about the Mahonian distribution.
Added Feb. 10, 2012: Persi Diaconis pointed out that we were scooped by him and other statisticians, a long time ago, see the erratum.

Sample Input and Output

  • Once you have the Maple package MahonianStat,
    • To get explicit, exact, formulas for the first twenty moments (about the mean) of the random variable "number of inversions" in the sample space of words with a 1's and b 2's, in terms of a and b,
      the input yields output.
    • To get the explicit formulas for the asymptotic expansion, with 4 terms, of the 2rth moment (about the mean) of the random variable "number of inversions" in the sample space of words with a 1's and b 2's, in terms of a and b, as a formula in a,b,t, and r(!)
      the input yields output.
    • To get the explicit formulas for the asymptotic expansion, with 7 terms, of the normalized 2rth moment ( α2r) (about the mean) of the random variable "number of inversions" in the sample space of words with a 1's and b 2's, in terms of a and b, as a formula in a,b,t, and r(!)
      the input yields output.
    • To get the asymptotic expression (with three terms) for the limiting distribution of the normalized random variable,
      the input yields output.
    Doron Zeilberger's List of Papers

    Doron Zeilberger's Home Page