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

  • Anna R.B. Fan
  • Harold R.L. Yang
  • Rebecca T. Yu
Keywords: partition, hook length, nearly $k$-triangular partition


Let $\alpha_k(\lambda)$ denote the number of $k$-hooks in a partition $\lambda$ and let $b(n,k)$ be the maximum value of $\alpha_k(\lambda)$ among partitions of $n$. Amdeberhan posed a conjecture on the generating function of $b(n,1)$. We give a proof of this conjecture. In general, we obtain a formula that can be used to determine $b(n,k)$. This leads to a generating function formula for $b(n,k)$. 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 $b(n,k)$.

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