Posted Oct. 5, 2010

The journal *Proceedings of the American Mathematical Society*,
erroneously (and, of course, stupidly) rejected
Andrew Baxter and Doron Zeilberger's article
The Number of Inversions and the Major Index of Permutations are Asymptotically Joint-Independently Normal,
based on a referee report that clearly showed that
he or she do not know how to read and understand a computer-assisted proof.
It is also clear that the referee thinks that a mathematically rigorous article has to be written
in the usual boring style. "Excited and chatty" does not mean non-rigorous.

Here is the report and Doron Zeilberger's responses.

Referee's report: >(1) Summary: >This paper outlines a method of proving that the pair permutation statistics >(inv,maj) are asymptotically joint-independently-normal. At the heart of this >is an interesting fast algorithm for computing the generating function for >the pair (inv,maj). The paper is written in a chatty and excited style, but is >painful to read since there are many missing details. I would very much like >to see this result proved in a conventional style. This article is written in a very clear and lucid way, outlining all theideas. It also describes where to find the corroborating computer software and its output that validate the arguments. This part has no place in the body of the article, and is described in the accompanying webpage, where all the claims are documented. >(2) Decision: >I do not recommend this paper for publication for the following reasons: >(i) The paper only gives an overview of the method of proof, so I cannot >vouch for its correctness. A slight effort to examine the accompanying Maple package, and the above webpage, should have convinced any Maple-literate person of the validity of the arguments. If the referee is not familiar with Maple, then he or she should have refused to review this paper. >(ii) Far too many aspects of this purported proof are essentially left as >"exercises to the reader". Everything is spelled-out, and I have no idea what the referee is talking about. Unlike the conventional boring style of mathematical discourse, it presents theideasvery clearly.

The associate editor who handled the submission agreed with the referee stating:

"Under its current policy, PAMS accepts papers that present mathematics supported by proofs that meet traditionally accepted rigorous standards."

As I said in Opinion 112, the present computer-assisted proof is far more rigorous than most human-generated proofs in the PAMS (or any other mathematics periodical, for that matter).

Added Oct. 27, 2010: This is not the first

Hi Doron, It probably was a bad decision. The essential problem I have as PAMS editor is that we have a rough quota of accepting 1 in 4 articles, and a lot of good submittals. I often get frustrated at dealing with so many articles and reject some of them for rather vacuous reasons. And, under this system, authors without a proven track record and/or who don't solve a known problem are at a disadvantage. So, the rejection is more a reflection of these conditions than a judgement about the math. Best, [Name removed]

Added Nov. 6, 2010: Watch the lecture .

Andrew Baxter and Doron Zeilberger's Article