7 research outputs found
On the Intriguing Problem of Counting (n+1,n+2)-Core Partitions into Odd Parts
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
Additional file 4: of Genetic variation and structure of maize populations from Saoura and Gourara oasis in Algerian Sahara
Figure S1. Principal component analysis of the 47 Algerian maize landraces based on 24 agro-morphological traits. (JPEG 152 kb
Additional file 1: of Genetic variation and structure of maize populations from Saoura and Gourara oasis in Algerian Sahara
Table S4. Augmented ANOVA (mean squares) for 24 agro-morphological traits in 47 Algerian maize landraces and 7 checks. (DOCX 17 kb
Additional file 6: of Genetic variation and structure of maize populations from Saoura and Gourara oasis in Algerian Sahara
Table S8. Analysis of Molecular Variance. (DOCX 12 kb
Additional file 7: of Genetic variation and structure of maize populations from Saoura and Gourara oasis in Algerian Sahara
Table S9. Partition of the gene diversity among clusters. (DOCX 12 kb
Additional file 9: of Genetic variation and structure of maize populations from Saoura and Gourara oasis in Algerian Sahara
Table S2. Agro-morphological traits recorded in the 47 Algerian maize landraces. (DOCX 15 kb
Additional file 2: of Genetic variation and structure of maize populations from Saoura and Gourara oasis in Algerian Sahara
Table S5. Correlation matrix of 24 agro-morphological traits used to characterize 47 Algerian maize landraces. (XLSX 16 kb