# Partitions and Sylvester waves

## TL;DR

This paper provides detailed descriptions and asymptotic analysis of Sylvester's waves, which decompose the restricted partition function, revealing when initial waves approximate the function well as both parameters grow large.

## Contribution

It offers the first asymptotic descriptions of the initial Sylvester waves for large N and n, enhancing understanding of partition function approximations.

## Key findings

- Asymptotic formulas for initial waves as N and n grow large
- Conditions under which initial waves approximate the partition function well
- Application of saddle-point method and dilogarithm in proofs

## Abstract

The restricted partition function $p_N(n)$ counts the partitions of the integer $n$ into at most $N$ parts. In the nineteenth century Sylvester described these partitions as a sum of waves. We give detailed descriptions of these waves and, for the first time, show the asymptotics of the initial waves as $N$ and $n$ both go to infinity at about the same rate. This allows us to see when the initial waves are a good approximation to $p_N(n)$ in this situation. Our proofs employ the saddle-point method of Perron and the dilogarithm.