Arithmetic of partitions and the $q$-bracket operator

TL;DR

This paper develops a multiplicative framework for integer partitions, revealing classical number theory results as special cases, and explores the deep role of the $q$-bracket operator in partition theory beyond modularity, including explicit formulas and inverse constructions.

Contribution

It introduces a multiplicative theory of partitions, connects classical number theory to combinatorial structures, and provides explicit formulas for the $q$-bracket operator and its inverse.

Findings

Classical number theory theorems are special cases of the new combinatorial laws.

Explicit formulas for the coefficients of the $q$-bracket operator are derived.

A partition-theoretic function is constructed with a prescribed $q$-bracket.

Abstract

We present a natural multiplicative theory of integer partitions (which are usually considered in terms of addition), and find many theorems of classical number theory arise as particular cases of extremely general combinatorial structure laws. We then see that the relatively recently-defined $q$-bracket operator $\left<f\right>_q$, studied by Bloch-Okounkov, Zagier, and others for its quasimodular properties, plays a deep role in the theory of partitions, quite apart from questions of modularity. Moreover, we give an explicit formula for the coefficients of $\left<f\right>_q$ for any function $f$ defined on partitions, and, conversely, give a partition-theoretic function whose $q$-bracket is a given power series.