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arXiv · 2207.03015

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

Sanjana Das
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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 $Λ_p$. Using his graph theoretic description of $Λ_p$, 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 $|Λ_p| \sim p^6/24$ as $p \to \infty$.

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Sanjana Das. 2022-08-21. Asymptotic Properties of Maximal $p$-Core $p'$-Partitions. https://arxiv.org/abs/2207.03015

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