conjugate {partitions}R Documentation

Conjugate partitions and Durfee squares

Description

Given a partition, provide its conjugate or Durfee square

Usage

conjugate(x)
durfee(x)

Arguments

x Either a vector describing a partition, in standard form (ie nonincreasing); or a matrix whose columns are partitions in standard form

Details

Conjugation is described in Andrews, and (eg) Hardy and Wright.

Essentially, conjugate() carries out R idiom rev(cumsum(table(factor(a[a>0],levels=max(a):1)))), but faster.

The “Durfee square” of a partition is defined on page 281 of Hardy and Wright. It is the largest square of nodes contained in the partition's Ferrers graph. Function durfee() returns the side of the Durfee square which Andrews denotes d(lambda). It is equivalent to R idiom function(a){sum(a>=1:length(a))}, but faster.

Value

Returns either a partition in standard form, or a matrix whose columns are partitions in standard form.

Note

If argument x is not nonincreasing, all bets are off: these functions will not work and will silently return garbage. Caveat emptor! (output from blockparts() is not necessarily non-increasing)

Author(s)

Robin K. S. Hankin

Examples

parts(5)
conjugate(parts(5))

restrictedparts(6,4)
conjugate(restrictedparts(6,4))

durfee(10:1)


[Package partitions version 1.9-3 Index]