On ℓ-regular bipartitions modulo ℓ

Let B ℓ ( n ) denote the number of ℓ -regular bipartitions of n . In this paper, we prove several infinite families of congruences satisfied by B ℓ ( n ) for ℓ ∈ { 5 , 7 , 13 } . For example, we show that for all α > 0 and n ≥ 0 , B 5 4 α n + 5 × 4 α - 2 6 ≡ 0 ( mod 5 ) , B 7 5 8 α n + 5 8 α - 1 2 ≡ 3 α B 7 ( n ) ( mod 7 ) and B 13 5 12 α n + 5 12 α - 1 ≡ B 13 ( n ) ( mod 13 ) .