TY - JOUR

T1 - Elementary proofs of parity results for 5-regular partitions

AU - Hirschhorn, Michael D.

AU - Sellers, James A.

PY - 2010/2

Y1 - 2010/2

N2 - 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, Integers 8 (2008), #A60] used the theory of modular forms to examine 5-regular partitions modulo2 and 13-regular partitions modulo2 and3; they obtained and conjectured various results. In this note, we use nothing more than Jacobis 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.

AB - 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, Integers 8 (2008), #A60] used the theory of modular forms to examine 5-regular partitions modulo2 and 13-regular partitions modulo2 and3; they obtained and conjectured various results. In this note, we use nothing more than Jacobis 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.

KW - Congruences

KW - Jacobis triple product identity

KW - Partitions

KW - Regular partitions

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U2 - 10.1017/S0004972709000525

DO - 10.1017/S0004972709000525

M3 - Article

AN - SCOPUS:77957231242

SN - 0004-9727

VL - 81

SP - 58

EP - 63

JO - Bulletin of the Australian Mathematical Society

JF - Bulletin of the Australian Mathematical Society

IS - 1

ER -