Congruences modulo 11 for broken 5-diamond partitions

• Published in: The Ramanujan journal 2018-05, Vol.46 (1), p.151-159
• Main Authors: Liu, Eric H., Sellers, James A., Xia, Ernest X. W.
• Format: Article
• Language: English
• Subjects: Combinatorics; Congruences; Diamonds; Field Theory and Polynomials; Fourier Analysis; Functions of a Complex Variable; Mathematics; Mathematics and Statistics; Number Theory; Partitions (mathematics)
• ISSN: 1382-4090; 1572-9303
• DOI: 10.1007/s11139-017-9894-5

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 ( mod 4 ) , there exists an integer λ ( p ) ∈ { 2 , 3 , 5 , 6 , 11 } such that, for n , α ≥ 0 , if p ∤ ( 2 n + 1 ) , then Δ 5 11 p λ ( p ) ( α + 1 ) - 1 n + 11 p λ ( p ) ( α + 1 ) - 1 + 1 2 ≡ 0 ( mod 11 ) . Moreover, some non-standard congruences modulo 11 for Δ 5 ( n ) are deduced. For example, we prove that, for α ≥ 0 , Δ 5 11 × 5 5 α + 1 2 ≡ 7 ( mod 11 ) .