The second minimal excludant and mex sequences

Abstract

The minimal excludant of an integer partition, first studied prominently by Andrews and Newman from a combinatorial viewpoint, is the smallest positive integer missing from a partition. Several generalizations of this concept are being explored by mathematicians nowadays. We analogously consider the second minimal excludant of a partition and analyze its relationship with the minimal excludant. This leads us to the notion of a mex sequence and we derive two neat identities involving the number of partitions whose mex sequence has length at least r