Written: Nov. x 2017

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, 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 sequnece' (see the article), is OEIS sequence A047749 that does have an algebraic generating function. We also develop algorithms, and find explicit generating functions, for other 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 simultatneous 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 sequnce of order-ideals of P_n,n+1 order-ideals whose vertices alternate in sign with labelling paramter c=0, and c=1, and their interleavigs, 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 order-ideals of P_{n+1,n+2} with alternating colors (with both c=0 and c=1) supported in the k out diagonals of the lattice triangle with "vertices" [0,0],[n-1,0],[0,n-1], for k=1,2,3,4

  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.

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