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.

Maple packages

• OddArmin.txt, a Maple package to enumerate (n+1,n+2)-core partitions with odd parts.

• core.txt, a Maple package to enumerate all kinds of simultaneous core partitions.

• stCorePlus.txt, a Maple package extending a previous Maple package to handle the objects counted here. It is mainly for checking purposes.

Sample Input and Output for OddArmin.txt

• If you want to see the first 23 terms of the sequence of the number of order-ideals of P_n,n+1 whose vertices alternate in "color" (parity) with labeling paramater c=0, and c=1, and their interleavings, giving sequence A047749 and our object of desires, the Armin Straub sequence enumerating the number of (n+1,n+2)-core partitions with odd parts,

  the input file generates the output file.

• If you want to see (rigorously proved) rational generating functions for the number of (n+1,n+2)-core partitions into odd parts, whose corresponding order-ideals of P_{n+1,n+2} are supported in the k outermost diagonals of the lattice triangle with "vertices" [0,0],[n-1,0],[0,n-1], for k=1,2,3,4,5

  the input file generates the output file.

• If you want to see (rigorously proved) rational generating functions for (n+1,n+2)-core-partitions where every part can get repeated at most k times, for k from 1 to 20

  the input file generates the output file.

• If you want to see (rigorously proved) rational generating functions for order-ideals of P_{n+1,n+2} supported in the k out diagonals of the lattice triangle with "vertices" [0,0],[n-1,0],[0,n-1], for k from 1 to 20

  the input file generates the output file.

Some Pictures of the Lattices and Order-Ideals

• The lattice P_9,10

• The lattice P_10,11

• 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

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