Partitions and the Minimal Excludant

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