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 (Delta _k(n)) denote the number of broken k-diamond partitions of n. Recently, Paule and Radu conjectured two relations on (Delta _5(n)) which were proved by Xiong and Jameson, respectively. In this paper, employing these relations, we prove that, for any prime p with (pequiv 1 (mathrm{mod} 4)), there exists an integer (lambda (p)in {2, 3, 5, 6, 11}) such that, for (n, alpha ge 0), if (pnot mid (2n+1)), then

$$begin{aligned} Delta _5left( 11p^{lambda (p)(alpha +1)-1} n+frac{11p^{lambda (p)(alpha +1)-1}+1}{2}right) equiv 0 (mathrm{mod} 11). end{aligned}$$

Moreover, some non-standard congruences modulo 11 for (Delta _5(n)) are deduced. For example, we prove that, for (alpha ge 0), (Delta _5left( frac{11times 5^{5alpha }+1}{2}right) equiv 7 (mathrm{mod} 11)).