Mappings and symmetries of partitions
type: Item
instance of: scholarly article
title: Mappings and symmetries of partitions
zbMATH Open document ID: 0682.05008
DOI: 10.1016/0097-3165(89)90043-5
author: Freeman Dyson
published in: Journal of Combinatorial Theory. Series A
publication date: 1989-01-01T00:00:00Z
review text: \textit{F. G. Garvan} [Trans. Am. Math. Soc. 305, No.1, 47-77 (1988; Zbl 0641.10009)] defined a vector partition of an integer n to be an ordered triple of ordinary partitions \(P=[P_ 1,P_ 2,P_ 3]\), where \(P_ 1\) is a partition into unequal parts and \(P_ 2\), \(P_ 3\) are unrestricted partitions into positive parts. The existence of a ``crank'' was conjectured by the author in 1944 and proved, along with various results concerning them, by \textit{G. Andrews} and \textit{F. G. Garvan} [Bull. Am. Math. Soc., N. Ser. 18, No.2, 167-171 (1988; Zbl 0646.10008)]. If p(k,n) is the number of partitions of n with crank k and \(N_ V(k,n)\) is the sum of the ``weights'' defined by Garvan, then Andrews and Garvan proved, using generating functions, that for all k and \(n>1\), \(p(k,n)=N_ V(k,n)\). The author provides a combinatorial interpretation of this result using a direct mapping of vector partitions onto ordinary partitions. He further defines an extended version of these vector partitions where \(P_ 2\) can have any number of zero parts, a ``rank set'' of the partition, and q(k,n), the number of partitions of n whose rank sets contain k. This allows an analogous result to be proved: \(q(k,n)=N_ E(k,n)\), for all n and k, where \(N_ E(k,n)\) is the sum of weights taken over all extended vector partitions of sum n, rank k. Further, \(q(k,n)=q(-k,n+k)\) and \(q(k,n)+q(-k-1,n)=p(n)\). Finally, the author shows that \(\sum_{k}k^ 2p(k,n)=2np(n)\), \(n>1\).
Mathematics Subject Classification ID: 05A17
Mathematics Subject Classification ID: 11P81
zbMATH DE Number: 4116539
zbMATH Keywords: vector partition
reviewed by: Gerald L. Alexanderson
MaRDI profile type: Publication
full work available at URL: 0097-3165(89)90043-5
OpenAlex ID: W2023497974
cites work: CONGRUENCES FOR ANDREWS' SMALLEST PARTS PARTITION FUNCTION AND NEW CONGRUENCES FOR DYSON'S RANK
cites work: New Combinatorial Interpretations of Ramanujan's Partition Congruences Mod 5,7 and 11
cites work: Dyson’s crank of a partition
cites work: A new symmetry of partitions
cites work: On the Geometry of Numbers in Elementary Number Theory
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description: scientific article; zbMATH DE number 4116539