parts {partitions} | R Documentation |
Given an integer, return a matrix whose columns enumerate its partitions.
Function parts()
returns the unrestricted partitions; function
diffparts()
returns the unequal partitions; function
restrictedparts()
returns the restricted partitions; function
blockparts()
returns the partitions subject to specified
maxima; and function compositions()
returns all compositions
of the argument.
parts(n) diffparts(n) restrictedparts(n, m, include.zero=TRUE, decreasing=TRUE) blockparts(f, n=NULL, include.fewer=FALSE) compositions(n, m=NULL, include.zero=TRUE)
n |
Integer to be partitioned. In function blockparts() ,
the default of NULL means to return all partitions of any size |
m |
In functions restrictedparts() and
compositions() , the order of the partition |
include.zero |
In functions restrictedparts() and
compositions() , Boolean with default FALSE meaning to
include only partitions of n into exactly m
parts; and TRUE meaning to include partitions of n into
at most m parts (because zero parts are included) |
include.fewer |
In function blockparts() , Boolean with
default FALSE meaning to return vectors whose sum is
exactly n and TRUE meaning to return partitions
whose sum is at most n |
decreasing |
In restrictedparts() , Boolean with default
TRUE meaning to return partitions whose parts are in
decreasing order and FALSE meaning to return partitions in
lexicographical order, as appearing in Hindenburg's
algorithm. Note that setting to decreasing to FALSE
has the effect of making conjugate() return garbage |
f |
In function blockparts() , a vector of strictly
positive integers that gives the maximal number of blocks; see
details |
parts()
uses the algorithm in Andrews.
Function diffparts()
uses a very similar algorithm that I
have not seen elsewhere. These functions behave strangely if given
an argument of zero.
restrictedparts()
uses the algorithm in
Andrews, originally due to Hindenburg. For partitions into at most
m parts, the same Hindenburg's algorithm is used but with a
start vector of c(rep(0,m-1),n)
.
blockparts()
enumerates the compositions of an
integer subject to a maximum criterion: given vector
y=(y_1,...,y_p) all sets of
a=(a_1,...,a_p) satisfying
sum(a_i)=n subject to 0<a_i<y_i for all i are given in lexicographical order.
If argument y
includes zero elements, these are treated
consistently (ie a position with zero capacity).
If n
takes its default value of NULL
, then
sum(a_i)=n is removed (the numbers may sum
to anything). Note that these solutions are not necessarily in
standard form, so functions durfee()
and conjugate()
may fail.
compositions()
returns all
2^(n-1) ways of partitioning an integer; thus
4+1+1
is distinct from 1+4+1
or 1+1+4
. This
function is different from all the others in the package in that it
is written in R; it is not clear that C would be any faster.
These vectorized functions return a matrix whose columns are the
partitions. If this matrix is too large, consider enumerating the
partitions individually using the functionality documented in
nextpart.Rd
.
Robin K. S. Hankin
parts(5) diffparts(10) restrictedparts(9,4) restrictedparts(9,4,FALSE) restrictedparts(9,4,decreasing=TRUE) blockparts(1:4) blockparts(1:4,3) blockparts(1:4,3,include.fewer=TRUE) blockparts(c(4,3,3,2),5) # Knuth's example, Fascicle 3a, p16 compositions(3)