Partitions and the Maximal Excludant - Chern, Shane (The Electronic Journal of Combinatorics, 2021)
For each nonempty integer partition π, we define the maximal excludant of π as the largest nonnegative integer smaller than the largest part of π that is not itself a part. Let σ\!maex(n) be the sum of maximal excludants over all partitions of n. We show that the generating function of σ\!maex(n) is closely related to a mock theta function studied by Andrews, Dyson and Hickerson, and Cohen, respectively. Further, we show that, as n→ ∞, σ\!maex(n) is asymptotic to the sum of largest parts over all partitions of n. Finally, the expectation of the difference of the largest part and the maximal excludant over all partitions of n is shown to converge to 1 as n→ ∞.