# On the Maximum Number of k-Hooks of Partitions of n

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Let $\alpha k\lambda$ denote the number of $k$-hooks in a partition $\lambda$ and let $bn,k$ be the maximum value of $\alpha k\lambda$ among partitions of $n$. Amdeberhan posed a conjecture on the generating function of $bn,1$. We give a proof of this conjecture. In general, we obtain a formula that can be used to determine $bn,k$. This leads to a generating function formula for $bn,k$. We introduce the notion of nearly $k$-triangular partitions. We show that for any $n$, there is a nearly $k$-triangular partition which can be transformed into a partition of $n$ that attains the maximum number of $k$-hooks. The operations for the transformation enable us to compute the number $bn,k$.

Author: Anna R. B. Fan; Harold R. L. Yang; Rebecca T. Yu

Source: https://archive.org/