# Complementary Schur Asymptotics for Partitions

## TL;DR

This paper derives asymptotic formulas for the number of partitions avoiding certain parts, extending classical results and constructing complex partition ideals.

## Contribution

It introduces asymptotics for restricted partition functions based on Hardy-Ramanujan results, enabling the construction of oscillating partition ideals.

## Key findings

- Derived asymptotics for partitions excluding finite sets of parts
- Extended classical Hardy-Ramanujan asymptotics to restricted partitions
- Constructed examples of highly oscillating partition ideals

## Abstract

We deduce from the strong form of the Hardy--Ramanujan asymptotics for the partition function $p(n)$ an asymptotics for $p_{-S}(n)$, the number of partitions of $n$ that do not use parts from a finite set $S$ of positive integers. We apply this to construct highly oscillating partition ideals.