New parity results for broken 11-diamond partitions

Abstract

The notation of broken k-diamond partitions was introduced in 2007 by Andrews and Paule. For a fixed positive integer k, let Δk(n) denote the number of broken k-diamond partitions of n. Recently, Radu and Sellers established numerous congruence properties for (2k+1)-cores by using the theory of modular forms, where k=2,3,5,6,8,9,11. Employing their congruences for (2k+1)-cores, Radu and Sellers obtained a number of nice parity results for Δk(n). In particular, they proved that for n⩾0, Δ11(46n+r)≡0(mod2), where r∈{11,15,21,23,29,31,35,39,41,43,45}. In this paper, we derive several new infinite families of congruences modulo 2 for Δ11(n) by using an identity given by Chan and Toh, and the p-dissection of Ramanujan's theta function f1 due to Cui and Gu. For example, we prove that for n⩾0 and k,α⩾1, Δ11(23α−2×23kn+23α−2s×23k−1+1)≡0(mod2), where s∈{5,7,10,11,14,15,17,19,20,21,22}. This generalizes the parity results for Δ11(n) discovered by Radu and Sellers.