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.
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 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 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)
# 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).