On the maximum number of \(k\)-hooks of partitions of \(n\) (Q396759)

Summary: 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 \(b(n,k)\).