Tewodros Amdeberhan and Armin Straub initiated the study of enumerating
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 number 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, 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, which
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, more tractable, families of (n+1,n+2)-core partitions.Comment: 12 pages, accompanied by Maple package. This version announces that
our questions were all answered by Paul Johnson, and a donation to the OEIS,
in his honor, has been mad