Bernard L. S. Lin

NEW RAMANUJAN TYPE CONGRUENCE MODULO 7 FOR 4-COLORED GENERALIZED FROBENIUS PARTITIONS

In this brief note, we prove one unexpected Ramanujan type congruence modulo 7 for the number cϕ4(n) of generalized Frobenius partitions of n with four colors.

GENERALISATION OF KEITH’S CONJECTURE ON 9-REGULAR PARTITIONS AND 3-CORES

Recently, Keith used the theory of modular forms to study 9-regular partitions modulo 2 and 3. He obtained one infinite family of congruences modulo 3, and meanwhile proposed one analogous conjecture. In this note, we show that 9-regular partitions and 3-cores satisfy the same congruences modulo 3. Thus, we first derive several results on 3-cores, and then generalise Keiths conjecture and get a stronger result, which implies that all of Keiths results on congruences modulo 3 are consequences of our result. DOI: 10.1017/S0004972714000343

ON THE EXPANSION OF A CONTINUED FRACTION OF ORDER TWELVE

In this paper, we study 2-, 3-, 4-, 6- and 12-dissections of a continued fraction of order twelve. Our main result is that when the aforementioned continued fraction and its reciprocal are expanded as power series, the sign of the coefficients is periodic, with period 12.

Extensive parity analysis of broken 5-diamond partitions

In this paper, we investigate congruences for Δ5(n) modulo 2 and characterize the parity of Δ5(4n + 1) and Δ5(4n + 2) according to the arithmetic property of n. As a consequence, we obtain various Ramanujan type congruences for Δ5(n). We also extend these results to several infinite families of congruences.

THE RESTRICTED 3-COLORED PARTITION FUNCTION MOD 3

In this paper, we investigate the divisibility of the function b(n), counting the number of certain restricted 3-colored partitions of n. We obtain one Ramanujan type identity, which implies that b(3n + 2) ≡ 0 (mod 3). Furthermore, we study the generating function for b(3n + 1) by modular forms. Finally, we find two cranks as combinatorial interpretations of the fact that b(3n + 2) is divisible by 3 for any n.