parts {partitions}R Documentation

Enumerate the partitions of an integer

Description

Given an integer, return a matrix whose columns enumerate its partitions.

Function parts() returns the unrestricted partions; 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.

Usage

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)

Arguments

n Integer to be partitioned. In function blockparts(), the default of NULL means to return all partitions of any size
m In restrictedparts(), 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

Details

Function 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.

Function 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).

Function 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 subect 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.

Function compositions() returns all 2^n 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.

Author(s)

Robin K. S. Hankin

References

G. E. Andrews. “The Theory of Partitions”, Cambridge University Press, 1998

Examples

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)

compositions(3)


[Package partitions version 1.7-4 Index]