conjugate {partitions}

Conjugate partitions and Durfee squares

Description

Given a partition, provide its conjugate or Durfee square

Usage

conjugate(x, sorted = TRUE)
durfee(x, sorted = TRUE)
durfee_sorted(x)

Arguments

x

Either a vector describing a partition or a matrix whose columns are partitions

sorted

A logical indicating whether the data is already in standard form. That is to say, are the data within each column sorted in decreasing order?

Details

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

The conjugate of a partition may be calculated by taking its Ferrers diagram and considering the partition defined by columns instead of rows. This may be visualised by flipping the Ferrers diagram about the leading diagonal.

Essentially, conjugate() carries out R idiom

rev(cumsum(table(factor(a[a>0],levels=max(a):1))))

but is 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 length of 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 is 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 non-increasing, you must use the sorted = FALSE flag. Otherwise, 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)

# A partition in nonstandard form --- use `sorted = FALSE`
x <- parts(5)[sample(5),]
durfee(x, sorted = FALSE)
conjugate(x, sorted = FALSE)

# Suppose one wanted partitions of 8 with no part larger than 3:

conjugate(restrictedparts(8,3))

# (restrictedparts(8,3) splits 8 into at most 3 parts;
# so no part of the conjugate partition is larger than 3).