A “supernormal” partition statistic

Abstract

We study a bijective map 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. The supernorm is connected to a family of maps we define, which suggests the potential to apply techniques from partition theory to identify and prove multiplicative properties of integers. As an application of “supernormal” mappings (i.e., pertaining to the supernorm statistic), we prove an analogue of a formula of Kural-McDonald-Sah to give arithmetic densities of subsets of N instead of relative densities of subsets of P like previous formulas of this type; this builds on works of Alladi, Ono, Wagner, and the first and third authors. We then make a brief study of pertinent analytic aspects of the supernorm. Finally, using a table of “supernormal” additive-multiplicative correspondences, we conjecture Abelian-type formulas that specialize to our main theorem and other known results.