Given an integer, return a matrix whose columns enumerate various 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)
multiset(v,n=length(v))
mset(v)
multinomial(v)
allbinom(n,k)
Integer to be partitioned. In function blockparts()
,
the default of NULL
means to return all partitions of any size
In functions restrictedparts()
and
compositions()
, the order of the partition
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)
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
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
In function blockparts()
, a vector of strictly
positive integers that gives the maximal number of blocks; see
details
In function multiset()
, an integer vector representing
a multiset. Argument n
is the size of the sample to be
taken
In function allbinom()
, the size of the set to be
chosen; arguments match those of choose()
Robin K. S. Hankin
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)
.
Functions parts()
and restrictedparts()
overlap in
functionality. Note, however, that they can return identical
partitions but in a different order: parts(6)
and
restrictedparts(6,6)
for example.
If m>n, the partitions are padded with zeros.
Function blockparts()
enumerates the compositions of an
integer subject to a maximum criterion: given vector
y=(y_1,...,y_n)y=(y_1,...,y_p) all sets of
a=(a_1,...,a_n)a=(a_1,...,a_p) satisfying
_i=1^pa_i=nsum(a_i)=n subject to 0 a_i
y_i0 <= 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 the
restriction _i=1^pa_i=nsum(a_i)=n is relaxed (so that
the numbers may sum to anything). Note that these solutions are not
necessarily in standard form, so functions durfee()
and
conjugate()
may fail.
With a single argument, compositions(n)
returns
all 2^n-12^(n-1) ways of partitioning an integer; thus
4+1+1
is distinct from 1+4+1
or 1+1+4
.
With two arguments, compositions(n,m)
returns all
nonnegative solutions to x_1++x_m=nx_1+...+x_m=n.
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.
Function multiset()
returns all ways of ordering a
multiset (mset()
is a low-level helper function).
Function multinomial(v)
returns all ways of
partitioning a set into distinguishable boxes of capacities
v[1], v[2],...,v[n]
. The number of columns is given by the
multinomial coefficient v_i
v_1\,v_2\,...\,v_nomitted.
Function allbinom(n,k)
is provided for convenience; it
enumerates the ways of choosing \(k\) objects from n
.
G. E. Andrews. “The theory of partitions”, Cambridge University Press, 1998
R. K. S. Hankin 2006. “Additive integer partitions in R”. Journal of Statistical Software, Volume 16, code snippet 1
R. K. S. Hankin 2007. “Urn sampling without replacement: enumerative combinatorics in R”. Journal of Statistical Software, Volume 17, code snippet 1
R. K. S. Hankin 2007. “Set partitions in R”. Journal of Statistical Software, Volume 23, code snippet 2
N. J. A. Sloane, 2008, The On-Line Encyclopedia of Integer Sequences. Sequence A001700
D. Knuth, 2004. The art of computer programming, pre-fascicle 2B “Generating all permutations”
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