Christian Krattenthaler's Non-Anonymous Report on "The number of inversions and the major index of permutations are asymptotically joint-independently-normal" by Andrew Baxter and Doron Zeilberger

Posted Nov. 5, 2010

[Note by the authors: We thank Christian Krattenthaler for his insightful (and careful!) report. His report is based on the previous version We have incorporated all his suggestions in the current version.]

This is a lovely paper proving that the inversion statistic and the major statistic are asymptotically independent and tend to a 2-dimensional normal distribution. I read the whole paper and find everything correct and well described. The proof is actually very elegant. I have also tried the Maple package, everything works fine.

I have only minor comments and suggestions, which I list below.

I believe that you should mention the article
Garsia, A. M, Gessel, I., Permutation statistics and partitions. Adv. in Math. 31 (1979), no. 3, 288-305.
at some point. It provides a generating function formula for the inv-maj-polynomials. Whether this formula could be useful for approaching the asymptotic joint-independence-normality, this is certainly a different issue. Nevertheless, it should be mentioned that this formula exists (and who knows, it may be useful in this context).

p.2: MacMahon[M] -> MacMahon [M]

p.2: Foata[F] -> Foata [F]

p.3: proof[E] -> proof [E]

p.3. Footnote 2: I suppose that [GKP, Section 8.2] refers to the
    fixed-points-of-permutations computation?
    I would prefer if at least some of the "gory details" are given.
    Is it so clear that the denominators do always cancel?
    Probably, but only one has looked at it more closely, right?

p.4: Let's -> Let us

p.4: Note that when we chop off the last entry,
      -> Note that, when we chop off the last entry,

p.5, line below (RecF): It is too early to say that (RecF) enables you to
    compute F(n,i) for i $A=B$ if and only if $A-B=0$.

p.8, l.5,6 from bottom: FM(.,.)(n) -> FM(.,.)(n,i)
    (several times)

p.9, displayed formulas in the bottom:
    The parentheses could better adapted to what they enclose.
    The minus sign before -81 should be before the fraction.

Back to Andrew Baxter and Doron Zeilberger's refereed article