Asymptotic Properties of Maximal $p$-Core $p'$-Partitions

TL;DR

This paper investigates the asymptotic size of the largest $p$-core $p'$-partition, establishing bounds and showing it grows approximately as $p^6/24$ for large primes.

Contribution

It provides asymptotic bounds for the maximal size of $p$-core $p'$-partitions and confirms the growth rate as $p$ becomes large.

Findings

Maximal size of $ ext{Lambda}_p$ is approximately $p^6/24$ for large $p$.

Established bounds for $| ext{Lambda}_p|$ for $p > 10^6$.

Confirmed the asymptotic growth rate as $p o fty$.

Abstract

For primes $p$, we study the maximal possible size of a $p$-core $p'$-partition (a partition with no hook lengths or parts divisible by $p$). McDowell recently proved that the maximum is attained by a unique partition, say $\Lambda_p$. Using his graph theoretic description of $\Lambda_p$, we prove for $p > 10^6$ that \[\frac{1}{24}p^6 - p^5\sqrt{p} < |\Lambda_p| < \frac{1}{24}p^6 - \frac{1}{200}p^5\sqrt{p},\] which shows that $|\Lambda_p| \sim p^6/24$ as $p \to \infty$.