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

Abstract

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

Minor revisions to clarify exposition

References in corpus (1)

• Zeros in the Character Tables of Symmetric Groups with an $\ell$-Core Index