A "supernormal" partition statistic

We study a natural bijection from integer partitions to the prime factorizations of integers that we call the "supernorm" of a partition, in which the multiplicities of the parts of partitions are mapped to the multiplicities of prime factors of natural numbers. In addition to being inherently interesting, the supernorm statistic suggests the potential to apply techniques from partition theory to identify and prove multiplicative properties of integers. We prove an analogue of a formula of Kural-McDonald-Sah: we give arithmetic densities of subsets of $\mathbb N$ instead of natural densities in $\mathbb P$ like previous formulas of this type, as an application of the supernorm, building on works of Alladi, Ono, Wagner and the first and third authors. We then make an analytic study of related partition statistics; and suggest speculative applications based on the additive-multiplicative analogies we prove, to conjecture Abelian-type formulas. In an appendix, using these same analogies, we give a conjectural partition-theoretic model of prime gaps.