Asymptotics of reciprocal supernorm partition statistics

We consider two multiplicative statistics on the set of integer partitions: the norm of a partition, which is the product of its parts, and the supernorm of a partition, which is the product of the prime numbers pᵢ p i indexed by its parts i . We introduce and study new statistics that are sums of reciprocals of supernorms on three statistical ensembles of partitions, labelled by their size |λ |=n | λ | = n , their perimeter equaling n , and their largest part equaling n . We show that the cumulative statistics of the reciprocal supernorm for each of the three ensembles are asymptotic to eγ log n e γ log n as n → ∞ n → ∞ .