Two-colored generalized Frobenius partitions and minimal-excludant sums over bipartitions

Let ${\text{cpsi}}_{2,a}(n)$ denote the number of $(2,a)$-colored Frobenius partitions of weight $n$, where the two rows have prescribed length difference. We study the two cases $a=0$ and $a=1$ and connect them with minimal-excludant statistics on bipartitions. Let $\sigma {\text{mex}}_2(n)$ be the sum of the Lin–Liu bipartition minimal excludants over all bipartitions of $n$, and let $E_2(n)$ be the number of bipartitions whose two component minimal excludants are equal. For all $n\ge 0$, we give a combinatorial proof of $${\text{cpsi}}_{2,0}(n)=2\sigma {\text{mex}}_2(n)\text{and}{\text{cpsi}}_{2,1}(n)=2\sigma {\text{mex}}_2(n)-E_2(n)\mathrm{.}$$ These identities give direct combinatorial interpretations of two-colored Frobenius partition functions in terms of bipartition minimal-excludant sums.