On inequalities and linear relations for 7-core partitions
type: Item
instance of: scholarly article
title: On inequalities and linear relations for 7-core partitions
zbMATH Open document ID: 1235.11094
author: Byungchan Kim
published in: Discrete Mathematics
publication date: 2010-04-27T00:00:00Z
review text: A partition of the positive integer \(n\) is said to be a \(t\)-core one if there are no hook numbers that are multiples of \(t\). Let \(a_t(n)\) be the number of \(t\)-core partitions of \(n\). Using Ramanujan's modular equations \textit{N. D. Baruah} and \textit{B. C. Berndt} [J. Comb. Theory, Ser. A 114, No. 6, 1024--1045 (2007; Zbl 1206.11132)] have obtained the recurrence \(a_5(4n+3)=a_5(2n+1)+2a_5(n)\) and \textit{A. Berkovich} and \textit{H. Yesilyurt} [Discrete Math. 308, No. 22, 5246--5259 (2008; Zbl 1223.05010)] have proved the inequalities \(a_7(2n+2)\geq 2a_7(n)\) and \(a_7(4n+6)\geq 10a_7(n)\). In this present paper the author generalizes these results and proves conjectures of Berkovich and Yesilyurt. The proofs are based on the theory of modular forms.
Mathematics Subject Classification ID: 11P84
Mathematics Subject Classification ID: 05A17
Mathematics Subject Classification ID: 11F11
zbMATH DE Number: 5701943
zbMATH Keywords: 7-core partition
zbMATH Keywords: modular forms
zbMATH Keywords: Deligne's bound
reviewed by: Ljuben Mutafchiev
MaRDI profile type: Publication
full work available at URL: j.disc.2009.09.024
OpenAlex ID: W2062964749
cites work: Partition identities and Ramanujan's modular equations
cites work: New identities for 7-cores with prescribed BG-rank
cites work: On the representations of integers by the sextenary quadratic form \(x^2+y^2+z^2+7s^2+7t^2+7u^2\) and 7-cores
cites work: Cranks and t-cores
cites work: Some Congruences for Partitions that are p -Cores
cites work: Defect zero blocks for finite simple groups
cites work: Q3952291
cites work: Explicit bounds for the number of 𝑝-core partitions
cites work: Majorations Explicites Pour le Nombre de Diviseurs de N
cites work: Q4449110
DOI: 10.1016/J.DISC.2009.09.024
Recommended article: Arithmetic properties for 7-regular partition triples
Recommended article: Arithmetic properties of 7-regular partitions
Recommended article: On certain weighted 7-colored partitions
Recommended article: A generalization of the Farkas and Kra partition theorem for modulus 7
Recommended article: A THEOREM ON SEVEN-COLORED OVERPARTITIONS AND ITS APPLICATIONS
Recommended article: A note on the parity of broken 7-diamond partitions
Recommended article: Linear inequalities concerning partitions into distinct parts
Recommended article: Q4304446
Recommended article: Parity results for 7-regular and 23-regular partitions
Recommended article: New congruences modulo powers of 2 for broken 3-diamond partitions and 7-core partitions
description: scientific article; zbMATH DE number 5701943