Inequalities and asymptotics for hook lengths in ℓ-regular partitions and ℓ-distinct partitions

Abstract

In this article, we study hook lengths in ℓ-regular partitions and ℓ-distinct partitions. More precisely, we establish hook length inequalities between ℓ-regular partitions and ℓ-distinct partitions for hook lengths 2 and 3, by deriving asymptotic formulas for the total number of hooks of length t in both partition classes, for $t=1,2,3$. From these asymptotics, we show that the ratio of the total number of hooks of length t in ℓ-regular partitions to those in ℓ-distinct partitions tends to a constant that depends on ℓ and t. We also provide hook length inequalities within ℓ-regular partitions and within ℓ-distinct partitions.