setparts | R Documentation |
Enumeration of set partitions
setparts(x) listParts(x,do.set=FALSE) vec_to_set(vec) vec_to_eq(vec)
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 |
do.set |
Boolean, with |
vec |
An integer vector representing a set partition |
A partition of a set \mjeqnS=\left\lbrace 1,...,n\right\rbraceS=1,...,n is a family of sets \mjeqnT_1,...,T_kT1,...,Tk satisfying
i\neq j\longrightarrow T_i\cap T_j=\emptysetunion(Ti,Tj) empty if i != j
_i=1^kT_k=Sunion(T1,T2,...,Tk)=S
T_i\neq\emptysetTi not empty for \mjeqni=1,..., k1,...,k
The induced equivalence relation has \mjeqni\sim ji~j if
and only if i and j belong to the same partition.
Equivalence classes may be listed using listParts()
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])
.
You can change the print method by setting, eg,
option(separator="")
.
Functions vec_to_set()
and vec_to_eq()
are low-level
helper functions. These take an integer vector, typically a column of a
matrix produced by setparts()
and return their set
representation.
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.
The clue package by Kurt Hornik contains functionality for
partitions (specifically cl_meet()
and cl_join()
) which
might be useful. Option do.set
invokes functionality from the
sets package by Meyer et al.
Note carefully that setparts(c(2,1,1))
does not
enumerate the ways of placing four numbered balls in three boxes of
capacities 2,1,1. This is because there are two boxes of capacity 1,
and swapping the balls between these boxes gives the same set
partition (because sets are unordered). To do this, use
multinomial(c(a=2,b=1,c=1))
. See the setparts
vignette
for more details.
Luke G. West (C++
) and Robin K. S. Hankin (R);
listParts()
provided by Diana Tichy
R. K. S. Hankin 2006. Additive integer partitions in R. Journal of Statistical Software, Code Snippets 16(1)
R. K. S. Hankin 2007. “Set partitions in R”. Journal of Statistical Software, Volume 23, code snippet 2
Kurt Hornik (2017). clue: Cluster ensembles. R package version 0.3-53. https://CRAN.R-project.org/package=clue
Kurt Hornik (2005). A CLUE for CLUster Ensembles. Journal of Statistical Software 14/12. doi: 10.18637/jss.v014.i12
parts
, print.partition
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 listParts(jj) # The induced equivalence classes jj <- restrictedparts(6,3,TRUE) setparts(jj) # partitions of a set of 6 elements into ncol(setparts(jj)) # _exactly_ 3 sets; cf StirlingS2[6,3]==90 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
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