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

TL;DR

This paper proves a conjecture about the generating function of the maximum number of k-hooks in partitions of n, introduces nearly k-triangular partitions, and provides formulas to compute this maximum.

Contribution

It offers a proof of Amdeberhan's conjecture, derives a general formula for b(n,k), and introduces nearly k-triangular partitions to determine maximum k-hooks.

Findings

Proof of Amdeberhan's conjecture on generating functions.

Formula for calculating b(n,k).

Introduction of nearly k-triangular partitions for maximum k-hooks.

Abstract

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 $b(n,k)$.