Arithmetic properties and asymptotic formulae for $$\sigma _o\text {mex}(n)$$ and $$\sigma _e\text {mex}(n)$$

The minimal excludant of an integer partition is the least positive integer missing from the partition. Let \(\sigma _o\text {mex}(n)\) (resp., \(\sigma _e\text {mex}(n)\)) denote the sum of odd (resp., even) minimal excludants over all the partitions of n. Recently, Baruah et al. proved a few congruences for these partition functions modulo 4 and 8, and asked for asymptotic formulae for the same. In this article, we find Hardy-Ramanujan type asymptotic formulae for both \(\sigma _o\text {mex}(n)\) and \(\sigma _e\text {mex}(n)\). We also prove some infinite families of congruences for \(\sigma _o\text {mex}(n)\) and \(\sigma _e\text {mex}(n)\) modulo 4 and 8