2017/12/28 by Anthony Zaleski, Doron Zeilberger, Zaleski, Anthony +1
Computer Science · Mathematics · #05A15 #05A16 #05A17 #05E10 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1712.10072
openalex publication_date 2017/12/28 · openalex created_date 2022/09/02 · openalex updated_date 2026/07/28
Tewodros Amdeberhan and Armin Straub initiated the study of enumerating\nsubfamilies of the set of (s,t)-core partitions. While the enumeration of\n(n+1,n+2)-core partitions into distinct parts is relatively easy (in fact it\nequals the Fibonacci number Fn+2), the enumeration of (n+1,n+2)-core\npartitions into odd parts remains elusive.\n Straub computed the first eleven terms of that sequence, and asked for a\n"formula," or at least a fast way, to compute many terms. While we are unable\nto find a "fast" algorithm, we did manage to find a "faster" algorithm, which\nenabled us to compute 23 terms of this intriguing sequence. We strongly believe\nthat this sequence has an algebraic generating function, since a "sister\nsequence" (see the article), is OEIS sequence A047749 that does have an\nalgebraic generating function. One of us (DZ) is pledging a donation of 100\ndollars to the OEIS, in honor of the first person to generate sufficiently many\nterms to conjecture (and prove non-rigorously) an algebraic equation for the\ngenerating function of this sequence, and another 100 dollars for a rigorous\nproof of that conjecture.\n Finally, we also develop algorithms that find explicit generating functions\nfor other, more tractable, families of (n+1,n+2)-core partitions.\n