setparts {partitions}R Documentation

Set partitions

Description

Enumeration of set partitions

Usage

setparts(x)

Arguments

x If a vector of length 1, the size of the set to be partitioned. If a vector of length greater than 1, return all equivalence relations with equivalence classes with sizes of the elements of x. If a matrix, return all equivalence classes with sizes of the columns of x

Details

A partition of a set S={1,...,n} is a family of sets T1,...,Tk satisfying

The induced equivalence relation has i~j if and only if i and j belong to the same partition.

There are exactly fifteen ways to partition a set of four elements:
(1234)
(123)(4), (124)(3), (134)(2), (234)(1)
(12)(34), (13)(24), (14)(23)
(12)(3)(4), (13)(2)(4), (23)(1)(4), (24)(1)(3), (34)(1)(2)
(1)(2)(3)(4)

Note that (12)(3)(4) is the same partition as, for example, (3)(4)(21) as the equivalence relation is the same.

Consider partitions of a set S of five elements (named 1,2,3,4,5) with sizes 2,2,1. These may be enumerated as follows:

> u <- c(2,2,1)
> setparts(u)
                                  
[1,] 1 1 1 1 1 1 1 1 1 1 1 1 3 3 3
[2,] 2 2 3 1 1 1 2 2 3 2 2 3 1 1 1
[3,] 3 2 2 3 2 2 1 1 1 3 2 2 2 1 2
[4,] 2 3 2 2 3 2 3 2 2 1 1 1 2 2 1
[5,] 1 1 1 2 2 3 2 3 2 2 3 2 1 2 2

See how each column has two 1s, two 2s and one 3. This is because the first and second classes have size two, and the third has size one.

The first partition, x=c(1,2,3,2,1), is read “class 1 contains elements 1 and 5 (because the first and fifth element of x is 1); class 2 contains elements 2 and 4 (because the second and fourth element of x is 2); and class 3 contains element 3 (because the third element of x is 3)”. Formally, class i has elements which(x==u[i]).

Value

Returns a matrix each of whose columns show a set partition; an object of class "partition". Type ?print.partition to see how to change the options for printing.

Author(s)

Luke G. West (C++) and Robin K. S. Hankin (R)

References

See Also

parts, print.partition

Examples

setparts(4)                # all partitions of a set of 4 elements

setparts(c(3,3,2))         # all partitions of a set of 8 elements
                           # into sets of sizes 3,3,2.

jj <- restrictedparts(5,3)
setparts(jj)               # partitions of a set of 5 elements into
                           # at most 3 sets

setparts(conjugate(jj))    # partitions of a set of 5 elements into
                           # sets not exceeding 3 elements

setparts(diffparts(5))     # partitions of a set of 5 elements into
                           # sets of different sizes


[Package partitions version 1.9-3 Index]