By Anthony Zaleski and Doron Zeilberger
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 F_{n+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 input file generates the output file.
the input file generates the output file.
the input file generates the output file.
The lattice P_{9,10}
An example of an order-ideal of P_{10,11} 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 P_{10,11} 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