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

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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