Emilie Hogan's Non-Anonymous Report on the Maple package accompanying the article "The Number of Inversions and the Major Index of Permutations are Asymptotically Joint-Independently Normal" by A. Baxter and D. Zeilberger

Here are my comments on the maple code part of your paper. Everything except what I have listed here are correct (meaning they match up with what is written in the paper).

1. In FM8oo,ee,oe,eo and GM8oo,ee,oe,eo: you call FM8m(), but need to call FM8m(n,i).

2. In FM8ee,oe and GM8ee,oe you should add the argument i (now it is just FM8ee(n), GM8ee(n), FM8oe(n), GM8oe(n)).

3. In FM8oe and GM8oe: Something is wrong. You are missing the (r-1) that is in the paper. But when r=1 there is something more wrong. As it is written in InvMaj.txt the output of FM8oe is:

[[(9/2)*(2*i-n)^2/n^3, 27*(2*i-n)^2/n^3, (135/2)*(2*i-n)^2/n^3, 126*(2*i-n)^2/n^3], [1, 1, 1, 1], [2, 2, 2, 2], [3, 3, 3, 3]]

when (I think) it should be [[1,1,1,1],[1,1,1,1],[1,1,1,1],[1,1,1,1]]. For r=2,3,4 the difference is a factor of (r-1) which is in the paper but not in the program. I don't know how to fix the problem is for r=1. You can't put in the (r-1) because you get a division by 0 error.

In GM8oe you have the (r-1), so the output is:

[[-(1/8)*(2*i-n)^2, -(1/16)*n^3*(2*i-n)^2, -(25/1152)*n^6*(2*i-n)^2, -(245/31104)*n^9*(2*i-n)^2], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]]

when (I think) it should be [[0,0,0,0],[0,0,0,0],[0,0,0,0],[0,0,0,0]]. So for r=2,3,4 it is correct, but it is not correct for r=1. Of course, this could be the output that you expected, but it is not the output I expected after reading the paper and InvMaj.txt, and looking at the output for FM8ee/eo/oo and GM8ee/eo/oo.

4. Procedure Gng seems to have no use in the rest of the package. It is the same as procedure G except it is missing 1/(n-1) in two places.

5. I don't see the use for A(n,p,q), Bd(n,p,q), Bng(n,p,q), or B(n,p,q). They aren't used by other procedures, they are not in the output portion of the website, and they are not mentioned in the paper.

6. I don't know what FindLeading is supposed to do, but it isn't used in any other procedures and it isn't in any of the input/output examples on the website. I also don't know what Bdokee is supposed to do, but again it isn't used in any other procedures and it isn't in any of the input/output examples on the website. If you keep them in InvMaj.txt add a description so that the user knows what they are supposed to be doing.

7. For the procedure Nor(L,i,n) it may be helpful to add a suggested set of inputs. I'm not sure how one would use this. Though it is not used in the rest of the procedures or on the website.

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