Partitions and the Minimal Excludant

Abstract

Fraenkel and Peled have defined the minimal excludant or “${{\,\mathrm{mex}\,}}$” function on a set S of positive integers is the least positive integer not in S. For each integer partition $\pi$, we define ${{\,\mathrm{mex}\,}}(\pi )$ to be the least positive integer that is not a part of $\pi$. Define $\sigma {{\,\mathrm{mex}\,}}(n)$ to be the sum of ${{\,\mathrm{mex}\,}}(\pi )$ taken over all partitions of n. It will be shown that $\sigma {{\,\mathrm{mex}\,}}(n)$ is equal to the number of partitions of n into distinct parts with two colors. Finally the number of partitions $\pi$ of n with ${{\,\mathrm{mex}\,}}(\pi )$ odd is almost always even.