Combinatorial Proof of the Minimal Excludant Theorem

The minimal excludant of a partition $λ$, $\rm{mex}(λ)$, is the smallest positive integer that is not a part of $λ$. For a positive integer $n$, $σ\, \rm{mex}(n)$ denotes the sum of the minimal excludants of all partitions of $n$. Recently, Andrews and Newman obtained a new combinatorial interpretations for $σ\, \rm{mex}(n)$. They showed, using generating functions, that $σ\, \rm{mex}(n)$ 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 $σ\, \rm{mex}(n)$. We generalize this combinatorial interpretation to $σ_r\, \rm{mex}(n)$, the sum of least $r$-gaps in all partitions of $n$. The least $r$-gap of a partition $λ$ is the smallest positive integer that does not appear at least $r$ times as a part of $λ$.

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