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

Anna R.B. Fan¹ ; Harold R.L. Yang¹ ; Rebecca T. Yu¹

¹Nankai University

The electronic journal of combinatorics, Tome 20 (2013) no. 2

Résumé

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 $b(n,k)$.

Détail

DOI : 10.37236/2991

Classification : 05A15, 05A17
Mots-clés : partition, hook length, nearly \(k\)-triangular partition