Written By Andrew Baxter and Doron Zeilberger
Second Edition (Fully (non-anonymously!) refereed) Published: Feb. 4, 2011.
We have asked several people to openly review this article, for correctness, novelty, and if they wish of its "significance" (or lack of!) (but that's optional, history would be the ultimate judge in that respect, but some of the people that I have contacted, and kindly agreed to participate didn't only want to be "correctness/novelty" checkers) .
The first person we asked was Herbert Wilf who very promptly (Oct. 22, 2010) sent us the following general report, commenting on the paper as a whole.
In addition we have asked the following people, with the following division of labor:
Over-all organization and general methodology (assuming that the parts are right) :
Vince Vatter: Here is Vince Vatter's detailed and wise non-anoymous referee report on the general methodology and organization (posted Jan. 3, 2011)
the combinatorial part:
The probability part:
[Added March 28, 2018: Josh Swanson, a soon-to-graduate Ph.D. student of Sara Billey, met this challenge, and will get the prize as soon as his paper will be posted on the arxiv. When he does that, there would be a link to his paper here.]
[Added Feb. 19, 2019: Josh Swanson has just posted his beautiful article. The promised prize of $300 has been mailed to him.]
Emilie Hogan:(report promised by Jan. 1, 2011, and arrived Dec. 26, 2010): Here is Emilie Hogan's very careful and competent non-anoymous referee report on the Maple package (added Jan. 11, 2011: all Emilie's minor corrections (that concern procedures that are not needed for the proof) were made in the current version of InvMaj .)
Ilias Kotsireas:(report promised by Jan. 1, 2011, and arrived Dec. 28, 2010) Here is Ilias Kotsireas's systematic independent software testing of the Maple package
Added Nov. 4, 2011: Marko Thiel has kindly discovered a (minor!) error in one of the "hand-waving" arguments in the paper. He also kindly fixed it! See Marko Thiel's message
Added Dec. 5, 2013: Marko Thiel has just posted this beautiful generalization
Important: This article is accompanied by Maple package
If you want to see a (pre-computed, to save you time!) table of all mixed-factorial moments FM(r,s)(n,i) (about the mean) of the pair of random variables (inv,maj) defined over the set of permutations of {1, ..., n} that end in i (for symbolic(!) n and i) but numeric r and s with 1 ≤ r,s, ≤ 8, the input gives the output.
If you want to see a (pre-computed, to save you time!) table of all leading terms of the mixed-factorial moments FM(r,s)(n,i) (about the mean) of the pair of random variables (inv,maj) defined over the set of permutations of {1, ..., n} that end in i (for symbolic(!) n and i) but numeric r and s with 1 ≤ r,s, ≤ 8, the input gives the output.
If you want to see the first 20 generating functions (weight-enumerators) for the pair (inv,maj) defined over permutations, the input gives the output.
If you want to get FM(1,1)(n,i) ab initio the input gives the output.
If you want to see the (beginnings, up to order 5) of the infinite-dimensional operators annihilating
FM(r,s)(n,i) mentioned on p. 10 of the article (2nd ed.), and given there only to first-order, the
input
gives the output.
[the page numberings in the input and output files refer to the first edition]
If you want to see the empirical-yet-rigorous proof of the explicit expressions for
the leading terms of the mixed-factorial moments
FM(r,s)(n,i) mentioned on p. 9 of the article (2nd ed.), equation (RecG')
the
input
gives the output.
[the page numberings in the input and output files refer to the first edition]
If you want to see the empirical-yet-rigorous proof of the explicit expressions for
the leading terms of the mixed-factorial moments
FM(r,s)(n,i) mentioned on p. 9 of the article (2nd ed.), equation (Gnn')
the
input
gives the output.
[the page numberings in the input and output files refer to the first edition]
Previous version:
Jan. 27, 2011. Posting
Guoniu Han's report
(Previous)^{2} version:
Jan. 13, 2011. Posting
Mireille Bousquet-Mélou's report.
(Previous)^{3} version: Jan. 10, 2011 (incorporating Emilie Hogan's suggestions).
(Previous)^{4} version of this webpage (NOT of article): Jan. 3, 2011. (Posting the reports of Vince Vatter, Emilie Hogan, and Ilias Kotsireas. Vince's suggestions will be incroporated once all the reports would arrive, Emilie's suggestions were already made);
(Previous)^{5} Dec. 14, 2010, posting Svante Janson's non-anoymous referee report on the probablistic part. His minor suggested changes (α should be a) would be incorporated once all the other reports arrive.
(Previous)^{6} Version (NOT of article): Nov. 30, 2010, posting Dan Romik's non-anoymous referee report on the probablistic part. His suggested changes (those that we agree with), would be incorporated once all the other reports arrive.
(Previous)^{7} Version: Nov. 5, 2011, incorporating the Christian Krattenthaler's careful and insightful non-anoymous referee report.