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 zillion^{2} 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