Combinatorial formulas for arithmetic density

TL;DR

This paper introduces a power series involving integer partitions and compositions that converges to the reciprocal of the arithmetic density of a subset of natural numbers as the parameter approaches 1.

Contribution

It presents a novel power series formula connecting arithmetic density with partition and composition structures, providing a new analytical tool.

Findings

Derived a power series for 1/d_S as q approaches 1

Connected arithmetic density to partition and composition coefficients

Provides a new analytical approach to density calculations

Abstract

Let $d_S$ denote the arithmetic density of a subset $S \subseteq \mathbb N$. We derive a power series in $q\in \mathbb C$, $|q|<1$, with co\"efficients related to integer partitions and integer compositions, that yields $1/d_S$ in the limit as $q\to 1$ radially.