Explicit Expressions for the Variance and Higher Moments of the Size of a Simultaneous Core Partition and its Limiting Distribution

Shalosh B. Ekhad and Doron Zeilberger

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Posted: Aug. 30, 2015.

This version: Sept. 1, 2015.

[Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger, as well as in arxiv.org.]

Dedicated to William Y.C. "Bill" Chen, the tireless apostle of enumerative and algebraic combinatorics in China (and beyond)

UPDATE ADDED March 31, 2020: Chaim Even-Zohar brilliantly proved the first challenge of our paper. A donation of $100 to the OEIS Foundation, in his honor, has been made.

UPDATE ADDED Sept. 1, 2015: See Marko Thiel and Nathan Williams very interesting comments where they point out that the Second Challenge (except for the upper bound of "two pages") was already present in Paul Johnson's article, and our Theorems 2 and 3 (that are now "official" theorems) are given in their recent paper.

A donation of $100 to the OEIS Foundation, in honor of Paul Johnson, Marko Thiel, and Nathan Williams, has been made.

Note that, in particular, all the theorems in our paper are now "real" theorems, even for ultra-orthodox mathematicians (like George Andrews and Christian Krattenthaler).

Jaclyn Anderson proved that if s and t are relatively prime positive integers, then there are only finitely many partitions whose set of hook-lengths (see wiki article) is disjoint from the set {s,t}. In fact she proved that there are (s+t-1)!/(s!t!) of them. Drew Armstrong (see here) conjectured a beautiful expression for the average size, namely

(s-1)(t-1)(s+t+1)/24   ,

and this has been recently (rather painfully) proved by Paul Johnson and reproved by Victor Y. Wang.

But the average is just the first question one can ask about a probability distribution. In the present article, we state absolutely certain expressions (but "officially" still conjectures) for the variance (showing in particular that is is rather large, and there is no "concentration about the mean"), and the third through the sixth moments. For the special case of (s,s+1)-core partitions, we go all the way to the 9th moment.
[Of course, with bigger computers, and better coding, one would be able to go further, but enough is enough.]
We also pose two challenges, and will be glad to donate 100 dollars each, to the OEIS foundation in honor of the first provers, regarding a "soft" and "global", yet rigorous, justification of our empirical approach, and for proving an intriguing conjecture about the limiting distribution.

Maple Package

Some Input and Output files for the Maple package stCore

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