partitions (version 1.8-2)

conjugate: 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

Value

• Returns either a partition in standard form, 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.

Examples

parts(5)
conjugate(parts(5))

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

durfee(10:1)