A "supernormal" partition statistic

TL;DR

This paper introduces the 'supernorm', a bijective map linking integer partitions to prime factorizations, enabling new insights into multiplicative properties and densities of subsets of natural numbers.

Contribution

It defines the supernorm map, explores its analytic properties, and applies it to derive an arithmetic density formula analogous to existing natural density results.

Findings

Established a bijective map between partitions and prime factorizations.

Derived an arithmetic density formula for subsets of natural numbers.

Conjectured Abelian-type formulas based on supernormal correspondences.

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. We make a brief study of pertinent analytic aspects of the supernorm. Then, as an application of "supernorma"' 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 $\mathbb N$ instead of natural densities in $\mathbb P$ like previous formulas of this type; this builds on works of Alladi, Ono, Wagner, and the first and third authors. Finally, using a table of "supernormal" additive-multiplicative correspondences, we conjecture Abelian-type formulas that specialize to our main theorem and other known results.