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.
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.
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.
Previous version:
Jan. 27, 2011. Posting
Guoniu Han'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.
Refereed by:
Mireille Bousquet-Mélou, Guoniu Han, Emilie Hogan, Svante Janson, Ilias Kotsireas,
Christian Krattenthaler, Dan Romik, Vince Vatter, and Herbert Wilf
.pdf
.ps
.tex
This is the third edition, announcing that Joshua Swanson won the 300 dollars prize.
First Editon Written: April 7, 2010.
Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger
(and arxiv.org)
Current version of this webpage (AND article): Feb. 4, 2011, posting the long-awaited second edition,
incorporating the wise and insightful suggestions of nine world-class (non-anonymous!) expert referees.
All the reports are in! We have now writen a new edition, obtained in the links above.
Historians of mathematics may wish to also see the
first edition.
Added Oct. 5, 2010: DZ is hereby offering a reward of $1000 for the first person
to point out a serious error in the argument of this article, that would invalidate
its main result. See Doron Zeilberger's
Opinion 112
Ever since Don Knuth
asked
me (after being asked by Svante Janson)
about the asymptotic covariance of the Number of Inversions and
the Major Index, that
Shalosh B. Ekhad
answered
so brilliantly, I dreamed of proving the much stronger result that they are asymptotically
independent (we already know that they are both normal, thanks to my academic great-grandfather Willi Feller).
Finally, in collaboration with my brilliant human disciple, Andrew Baxter, and also with
the great help of Shalosh B. Ekhad (who was asked to be a coauthor, but gracefully declined),
we did it!
Non-Anonymous Referee Reports
[Note: We agree with all his remarks except that he misunderstood our claim that there is
"no closed form for the generating function". We meant H(n)(p,q) themselves not
the (weired-normalized generating function of Roselle mentioned by Knuth that Romik refers to).
While it is certainly possible that one can, "in principle", use this generating function to give another proof,
we doubt whether it would be easier, and we take issue with the dismissive tone implied
by "standard methods". DZ is hereby offering $300 for such a proof, not to exceed the length of the
current paper]. (In principle, the Riemann Hypothesis should be provable, using "standard techniques" of
analytic number theory).
Maple Package
Sample Input and Output for InvMaj
[the page numberings in the input and output files refer to the first edition]
[the page numberings in the input and output files refer to the first edition]
[the page numberings in the input and output files refer to the first edition]
Page History
(Previous)2 version:
Jan. 13, 2011. Posting
Mireille Bousquet-Mélou's report.
Doron Zeilberger's List of Papers