Robin K. S. Hankin

English

Research Papers in Economics

JSS Journal of Statistical Software
May 2006, Volume 16, Code Snippet 1. http://www.jstatsoft.org/
Additive Integer Partitions in R
Robin K. S. Hankin
National Oceanography Centre, Southampton
Abstract
This paper introduces the partitions package of R routines, for numerical calculation
of integer partititions. Functionality for unrestricted partitions, unequal partitions, and
restricted partitions is provided in a small package that accompanies this note; the em-
phasis is on terse, eﬃcient C code. A simple combinatorial problem is solved using the
package.
Keywords: integer partitions, restricted partitions, unequal partitions, R.
1. Introduction
A partition of a positive integer n is a non-increasing sequence of positive integers λ1,λ2,...,λr
such that
Pr
i=1 λi = n. The partition (λ1,...,λr) is denoted by λ, and we write λ ` n to
signify that λ is a partition of n. If, for 1 6 j 6 n, exactly fj elements of λ are equal to j,
we write λ =
 
1f1,2f2,...,nfn
; this notation emphasises the number of times a particular
integer occurs as a part. The standard reference is Andrews (1998).
The partition function p(n) is the number of distinct partitions of n. Thus, because
5 = 4 + 1 = 3 + 2 = 3 + 1 + 1 = 2 + 2 + 1 = 2 + 1 + 1 + 1 = 1 + 1 + 1 + 1 + 1
is a complete enumeration of the partitions of 5, p(5) = 7 (recall that order is unimportant:
a partition is deﬁned to be a non-increasing sequence).
Various restrictions on the nature of a partition are often considered. One is to require that
the λi are distinct; the number of such partitions is denoted q(n). Because
5 = 4 + 1 = 3 + 2
is the complete subset of partitions of 5 with no repetitions, q(5) = 3.
One may also require that n be split into exactly m parts. The number of partitions so
restricted may be denoted r(m,n).2 Additive Integer Partitions in R
2. Package partitions in use
The R (R Development Core Team 2006) package partitions associated with this paper may
be used to evaluate the above functions numerically, and to enumerate the partitions they
count. In the package, the number of partitions is given by P(), and the number of unequal
partitions by Q(). For example,
> P(100)
[1] 190569292
agreeing with the value given by Abramowitz and Stegun (1965). The unequal partitions of
an integer are enumerated by function diffparts():
> diffparts(10)
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
[1,] 10 9 8 7 7 6 6 5 5 4
[2,] 0 1 2 3 2 4 3 4 3 3
[3,] 0 0 0 0 1 0 1 1 2 2
[4,] 0 0 0 0 0 0 0 0 0 1
where the columns are the partitions. Finally, function restrictedpartitions() enumerates
the partitions of an integer into a speciﬁed number of parts.
2.1. A combinatorial example
Consider random sampling, with replacement, from an alphabet of a letters. How many draws
are required to give a 95% probability of choosing each letter at least once? I show below
how the partitions package may be used to answer this question exactly.
A little thought shows that the number of ways to draw each letter at least once in n draws
is
N =
X
λ`n;a
a!
Qr
i=1 fi!
·
n!
Qn
j=1 λj!
(1)
where the sum extends over partitions λ of n into exactly a parts; the ﬁrst term gives the
number of ways of assigning a partition to letters; the second gives the number of distinct
arrangements.
The corresponding R idiom is to deﬁne a nonce function f() that returns the product of
the two denominators, and to sum the requisite parts by applying f() over the appropriate
restricted partitions. The probability of getting all a letters in n draws is thus N/an, computed
by function prob():
> f <- function(x) {
+ prod(factorial(x), factorial(tabulate(x)))
+ }
> prob <- function(a, n) {Journal of Statistical Software – Code Snippets 3
+ jj <- restrictedparts(a, n)
+ N <- factorial(a) * factorial(n) * sum(1/apply(jj, 2, f))
+ return(N/a^n)
+ }
In the case of a = 4, we obtain n = 16 because prob(4,15) ' 0.947 and prob(4,16) ' 0.96.
3. Conclusions
The partitions package was developed to answer the combinatorial word question discussed
above: it does so using fast C code. Further work would include the enumeration of compo-
sitions and vector compositions.
Acknowledgements
I would like to acknowledge the many stimulating and helpful comments made by the R-help
list while preparing the package.
References
Abramowitz M, Stegun IA (1965). Handbook of Mathematical Functions. Dover, New York.
Andrews GE (1998). The Theory of Partitions. Cambridge University Press, Cambridge.
R Development Core Team (2006). R: A Language and Environment for Statistical Computing.
R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0, URL http:
//www.R-project.org/.
Aﬃliation:
Robin K. S. Hankin
National Oceanography Centre, Southampton
European Way
Southampton SO14 3ZH
United Kingdom
E-mail: r.hankin@noc.soton.ac.uk
URL: http://www.noc.soton.ac.uk/
Journal of Statistical Software http://www.jstatsoft.org/
published by the American Statistical Association http://www.amstat.org/
Volume 16, Code Snippet 1 Submitted: 2006-04-27
May 2006 Accepted: 2006-05-17

Additive Integer Partitions in R

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