# Vince Vatter's Non-Anonymous Report on the article "The Number of Inversions and the Major Index of Permutations are Asymptotically Joint-Independently Normal" by A. Baxter and D. Zeilberger

I greatly enjoyed reading your paper "The number of inversions and the major index of permutations are asymptotically joint-independently-normal". It is another great example of *rigorous* experimental mathematics.

All of the mathematics seems fine to me (although I must confess that I didn't check any of the algebra), so I have only minor comments and suggestions for you. I have tried not to duplicate the reports by Christian Krattenthaller and Dan Romik to keep things simple, and have instead remarked on the (few) places I got stuck, and on a few minor matters of style you might consider changing.

Page 1, Human Statistics: When I read that human statistics are "usually" joint-asymptotically normal, I must admit I was confused. First of all, you really mean to say that *pairs* of human statistics are usually joint-asymptotically normal. But then, I struggled to find an example which is not joint-asymptotically normal. I suppose by this you mean something silly like the pair {height, height + 1}?

Page 1, Permutation Statistics: What does it mean that Dominique Foata got his "3rd-cycle doctorate" in statistics?

Page 2, Inv: Instead of "it features in the definition of the determinant", you might simply say that "it determines the sign of the permutation".

Page 2, E[Inv]: I'm not sure that Feller would want credit for computing E[inv], since it is just a symmetry observation.

Page 3: "... Ekhad's brilliant answer for..." would, I think, read better as "... Ekhad's brilliant derivation of...".

Page 4: When you mention that there are closed-form expressions for G(n)(1,q) and G(n)(p,1), you might remind the reader that these two polynomials are the same.

Page 6, Second-to-last paragraph: Why does it matter that your finite sets are the symmetric groups?

Page 8, "Nice conjectures but what about proofs?", first paragraph: First, "write-down" should be "write down". More importantly, I really struggled to understand what you are attempting to outline here. You talk about expressing FM(r,s) in terms of FM(r',s') with r'+s' < r+s, which is fine with me. But then you point out that in order to compute FM(r,s), we need only finitely many terms of this recurrence. I expended considerable effort trying to figure out why this is not obvious, and came up empty handed. Once I read further, I figured out what you were doing, but this paragraph led me quite astray. As a minor point, when you list all four functions in this paragraph ("FM(2r,2s)(n,i), ..."), I don't see why; I think you might as well just say "FM(r,s)(n,i)". Also in that paragraph, am I correct that "depending explicitly on r and s as well as on n and i" is synonymous with "for symbolic r, s, n, and i"?

Page 8, Last paragraph: "Plug-in" should be "plug in", or if you really want to be fancy, "substitute".

Throughout: Ekhad is refered to as "Shalosh", "Shalosh B. Ekhad", "Ekhad", and "S. B. Ekhad" in various parts of the paper. For readers who don't know him personally, this might be confusing.

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