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

Let αk(λ) denote the number of k-hooks in a partition λ and let b(n, k) be the maximum value of αk(λ) 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).