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

Abstract

Let $\cpsi_{2,a}(n)$ 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 $σ\mex_2(n)$ 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 \[ \cpsi_{2,0}(n)=2σ\mex_2(n) \qquad\text{and}\qquad \cpsi_{2,1}(n)=2σ\mex_2(n)-E_2(n). \] These identities give direct combinatorial interpretations of two-colored Frobenius partition functions in terms of bipartition minimal-excludant sums.