Asymptotic Formula for $(t+1)$-Regular Partitions

TL;DR

This paper extends Hagis's asymptotic formula for the number of $(t+1)$-regular partitions of an integer, using the saddle point method to cover broader ranges of $t$ and providing explicit bounds, with applications to symmetric group character tables.

Contribution

It introduces a saddle point approach to generalize and refine asymptotic formulas for $p(N,t)$ across different ranges of $t$, including explicit bounds.

Findings

Extended asymptotic formulas for $p(N,t)$ using saddle point method.

Provided explicit bounds for the asymptotic estimates.

Applied results to estimate zeros in symmetric group character tables.

Abstract

A partition is $t$-regular if none of its parts is divisible by $t$. Let $p(N,t)$ be the number of $(t+1)$-regular partitions of a positive integer $N$. In 1971, Hagis proved an asymptotic formula for $p(N,t)$ using the circle method, when $t$ fixed. In this article, we use the saddle point method and extend the result of Hagis in different ranges of $t$, obtaining explicit bounds. We also discuss an application of our result to estimate zeros in the character table of the symmetric group.