By Anthony Zaleski and Doron Zeilberger
.pdf .ps .tex
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.
The lattice P9,10
An example of an order-ideal of P10,11 corresponding to an (10,11)-core partition into odd parts (in other words, the sorted list of labels of the occupied cells alternate in parity) can be seen here
Another example of an order-ideal of P10,11 corresponding to an (10,11)-core partition into odd parts (in other words, the sorted list of labels of the occupied cells alternate in parity) can be seen here
Doron Zeilberger's Home Page
Anthony Zaleski's Home Page