2018/02/26 by Paul D. Johnson, Johnson, Paul, Paul E. Johnson
Engineering · Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.1802.09621
openalex publication_date 2018/02/26 · openalex created_date 2022/08/29 · openalex updated_date 2026/07/28
IMPORTANT NOTE: This paper is much rougher than I'd usually submit, and not\nentirely complete, though the main theorems and proofs should not be hard to\nfollow. Given the ongoing strike at UK Universities it may be some time before\nI get to complete it to my satisfaction, and in the meantime people I've shared\nthe preliminary draft with would like to be able to reference it. Hence I'm\nuploading it in its current form, and will update it later.\n The main new result of this paper is to count the number of (n,n+1)-core\npartitions with odd parts, answering a question of Zaleski and Zeilberger with\nbounty a charitable contribution to the OEIS. Along the way, we prove a general\ntheorem giving a recurrence for (n,n+1)-core parts whose smallest part and\nconsecutive part differences are restricted to lie in an arbitrary set M. This\ntheorem unifies many known results about (n,n+1)-core partitions with\nrestrictions.\n We end with discussions of extensions of the general theorem that keep track\nof the largest part, number of parts, and size of the partition, and about a\nfew cases where the same methods work on more general simultaneous cores.\n