ELEMENTARY PROOFS OF PARITY RESULTS FOR 5-REGULAR PARTITIONS
By Michael D. Hirschhorn, James A. Sellers. Bulletin of the Australian Mathematical Society, 2009
126 citations recorded by the source corpus.
Abstract In a recent paper, Calkin et al. [N. Calkin, N. Drake, K. James, S. Law, P. Lee, D. Penniston and J. Radder, ‘Divisibility properties of the 5-regular and 13-regular partition functions’, Integers8 (2008), #A60] used the theory of modular forms to examine 5-regular partitions modulo 2 and 13-regular partitions modulo 2 and 3; they obtained and conjectured various results. In this note, we use nothing more than Jacobi’s triple product identity to obtain results for 5-regular partitions that are stronger than those obtained by Calkin and his collaborators. We find infinitely many Ramanujan-type congruences for b5(n), and we prove the striking result that the number of 5-regular partitions of the number n is even for at least 75% of the positive integers n.
Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics
View Paper: https://www.semanticscholar.org/paper/310e440faa66ee9c9d30d56528ffca27c6d6e855
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In Algebra and Number Theory
In Discrete Mathematics and Combinatorics
Citations in the corpus
- Congruences for 9-regular partitions modulo 3 (2013, background): “In recent years there has been a spate of work ([1], [8], [5], [7], [6] and more) on the divisibility properties of bl(n) in arithmetic progression for various l, especially on their parity.”