On $$ell $$-regular bipartitions modulo $$ell $$

Let (B_ell (n)) denote the number of (ell )-regular bipartitions of n. In this paper, we prove several infinite families of congruences satisfied by (B_ell (n)) for (ell in {{5,7,13}}). For example, we show that for all (alpha >0) and (nge 0),

$$begin{aligned} B_5left( 4^alpha n+frac{5times 4^alpha -2}{6}right)equiv & {} 0 (text {mod} 5), B_7left( 5^{8alpha }n+displaystyle frac{5^{8alpha }-1}{2}right)equiv & {} 3^alpha B_7(n) (text {mod} 7) end{aligned}$$

and

$$begin{aligned} B_{13}left( 5^{12alpha }n+5^{12alpha }-1right) equiv B_{13}(n) (text {mod} 13). end{aligned}$$