Asymptotic Properties of Maximal $p$-Core $p^{\mathrm{\prime }}$-Partitions

For primes $p$, we study the maximal possible size of a $p$-core $p^{\mathrm{\prime }}$-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 ${\mathrm{\Lambda }}_p$. Using his graph theoretic description of ${\mathrm{\Lambda }}_p$, we prove for $p>{10}^6$ that $$\frac{1}{24}{p}^6-p^5\sqrt{p}<\mathrm{\mid }{\mathrm{\Lambda }}_p\mathrm{\mid }<\frac{1}{24}{p}^6-\frac{1}{200}{p}^5\sqrt{p},$$ which shows that $\mathrm{\mid }{\mathrm{\Lambda }}_p\mathrm{\mid }\sim p^6\mathrm{/}24$ as $p\to \mathrm{\infty }$.