Combinatorial Proof of the Minimal Excludant Theorem

The minimal excludant of a partition $\lambda$, $\rm{mex}(\lambda)$, is the smallest positive integer that is not a part of $\lambda$. For a positive integer $n$, $\sigma\, \rm{mex}(n)$ denotes the sum of the minimal excludants of all partitions of $n$. Recently, Andrews and Newman obtained a new combinatorial interpretations for $\sigma\, \rm{mex}(n)$. They showed, using generating functions, that $\sigma\, \rm{mex}(n)$ equals the number of partitions of $n$ into distinct parts using two colors. In this paper, we provide a purely combinatorial proof of this result and new properties of the function $\sigma\, \rm{mex}(n)$. We generalize this combinatorial interpretation to $\sigma_r\, \rm{mex}(n)$, the sum of least $r$-gaps in all partitions of $n$. The least $r$-gap of a partition $\lambda$ is the smallest positive integer that does not appear at least $r$ times as a part of $\lambda$.