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

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 $p \to \infty$.