Generalization of Dyson's rank and non-Rogers-Ramanujan partitions
type: Item
instance of: scholarly article
title: Generalization of Dyson's rank and non-Rogers-Ramanujan partitions
zbMATH Open document ID: 0819.11043
DOI: 10.1007/BF02567461
author: Frank G. Garvan
published in: Manuscripta Mathematica
publication date: 1995-01-25T00:00:00Z
full work available at URL: https://eudml.org/doc/155995
review text: The rank of a partition is a statistic that splits the set of partitions of \(5n+ 4\) and \(7n+ 5\) into, respectively, five and seven equinumerous sets, thus proving that \(p(5n+ 4)\equiv 0\pmod 5\) and \(p(7n+ 5)\equiv 0\pmod 7\). The crank is a similar statistic that separates the partitions of \(11n+ 7\) into eleven equinumerous sets. The author describes a \(k\)- rank for which the rank is the 2-rank and the crank is the 1-rank, and defines a generalized notion of conjugation, \(k\)-conjugation, which reverses the sign of the value of the \(k\)-rank. The author proves that the number of self-\(2k\)-conjugate partitions of \(n\) is equal to the number of partitions of \(n\) in which no part is divisible by \(2k\) and parts divisible by \(k\) are distnct.
Mathematics Subject Classification ID: 11P83
Mathematics Subject Classification ID: 05A17
zbMATH DE Number: 703888
zbMATH Keywords: partitions
zbMATH Keywords: \(k\)-rank
zbMATH Keywords: crank
zbMATH Keywords: \(k\)-conjugation
zbMATH Keywords: number of self-\(2k\)- conjugate partitions
reviewed by: David M. Bressoud
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 703888