Marko Thiel's and Nathan William's Feedback on S. B. Ekhad and D. Zeilberger's article "Explicit Expressions for the Variance and Higher Moments of the Size of a Simultaneous Core Partition and its Limiting Distribution"

Below are very interesting remarks by Marko Thiel and Nathan Williams about our article. Even though, strictly speaking, the "soft" argument that the moments are always polynomials far exceeds two pages (if one has to gather all the prerequisites), we gladly made a donation to the OEIS in their honor, as well as in honor of Paul Johnson.

Email Message from Marko Thiel (Sept. 1. 2015)

Dear Professor Zeilberger,

I saw your nice preprint on the moments and limiting distribution of the size of cores today.

I am happy to inform you that your second challenge has essentially already been met by Paul Johnson. In his paper, he interprets (a,b)-cores as lattice points in the b-th dilation of an (a-1)-dimensional simplex and size as a quadratic function. By Euler-Maclaurin theory, a generalization of Ehrhart theory, this implies that the sum of the m-th power of the size over cores is a polynomial of degree a-1+2m in b for every a.

The zero (multi-)set of this polynomial contains the zero (multi-)set of the (rational Catalan) polynomial in b that counts (a,b)-cores, so the average of the m-th power of the size of cores is a polynomial of degree 2m in b for every a. By symmetry, it is also a polynomial of degree 2m in a for every b. So Lagrange interpolation implies that it is a polynomial in both a and b. Its total degree is bounded by 4m.

For a more conceptual and uniform view of this story, see my recent preprint with Nathan Williams.

Kind Regards,

Marko Thiel

Email Message from Nathan Williams (Sept. 1, 2015)

Dear Doron

To be explicit, I would like to point out that our Theorem 1.5 is your Theorem 2, and our Theorem 1.6 is your Theorem 3. In Conjecture 8.5 we give conjectured leading coefficients (that is, in your notation, the polynomial in t for the highest power of s) for the polynomials up to the seventh moment (and one could certainly compute more, since they are given by relatively simple integrals).

Kind Regards,

Nathan Williams

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