On the maximum number of \(k\)-hooks of partitions of \(n\)
type: Item
instance of: scholarly article
title: On the maximum number of \(k\)-hooks of partitions of \(n\)
zbMATH Open document ID: 1295.05031
author: Anna R. B. Fan
author: Harold R. L. Yang
author: Rebecca T. Yu
published in: The Electronic Journal of Combinatorics
publication date: 2014-08-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/1212.3505
full work available at URL: v20i2p49
review text: 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 \(b(n,k)\).
Mathematics Subject Classification ID: 05A15
Mathematics Subject Classification ID: 05A17
zbMATH DE Number: 6330257
zbMATH Keywords: partition
zbMATH Keywords: hook length
zbMATH Keywords: nearly \(k\)-triangular partition
MaRDI profile type: Publication
arXiv ID: 1212.3505
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description: scientific article; zbMATH DE number 6330257