On the Intriguing Problem of Counting (n+1,n+2)-core partitions into Odd Parts

By Anthony Zaleski and Doron Zeilberger

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Exclusivley published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger and arxiv.org

First Written: Dec. 26, 2017

Last update: Feb. 28, 2018

UPDADTE Feb. 28, 2018: Paul Johnson has just posted his beautiful article that does much more than we asked. A donation of $200 to the OEIS Foundation was made.

Abstract: Tewodros Amdeberhan and Armin Straub initiated the study of subfamilies of the set of (s,t)-core partitions. While the enumeration of (n+1,n+2)-core partitions into distinct parts is relatively easy (in fact it equals the Fibonacci numbers Fn+2), the enumeration of (n+1,n+2)-core partitions into odd parts remains elusive. Straub computed the first eleven terms of that sequence, (see penultimate slide of Armin Straub's talk), and asked for a "formula", or at least a fast way, to compute many terms. While we are unable to find a "fast" algorithm, we did manage to find a "faster" algorithm, that enabled us to compute 23 terms of this intriguing sequence. We strongly believe that this sequence has an algebraic generating function, since a "sister sequence" (see the article), is OEIS sequence A047749 that does have an algebraic generating function. One of us (DZ) is pledging a donation of 100 dollars to the OEIS, in honor of the first person to generate sufficiently many terms to conjecture (and prove non-rigorously) an algebraic equation for the generating function of this sequence, and another 100 dollars for a rigorous proof of that conjecture.

Finally, we also develop algorithms that find explicit generating functions, for other, seemingly more tractable, families of (n+1,n+2)-core partitions.

Added Jan. 24, 2018: Paul Johnson noticed (and proved!) that the "sister sequence" is the number of (n+1,n+2)-core partitions into even parts, and even more impressively related these two sequences, completely solving the two challenges posed in this paper.(See above update)

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Sample Input and Output for OddArmin.txt

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