Congruences modulo 11 for broken 5-diamond partitions

The notion of broken k-diamond partitions was introduced by Andrews and Paule in 2007. For a fixed positive integer k, let Δ _k(n) denote the number of broken k-diamond partitions of n. Recently, Paule and Radu conjectured two relations on Δ ₅(n) which were proved by Xiong and Jameson, respectively. In this paper, employing these relations, we prove that, for any prime p with p≡1(mod4), there exists an integer λ(p)∈{2,3,5,6,11} such that, for n, α≥ 0 , if p∤ (2 n+ 1) , then (Folmula presented.).Moreover, some non-standard congruences modulo 11 for Δ ₅(n) are deduced. For example, we prove that, for α≥ 0 , Δ5(11×55α+12)≡7(mod11).