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 σ mex ( n ) to be the sum of mex ( π ) taken over all partitions of n . It will be shown that σ mex ( 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.