Given an integer, P() returns the number of additive
partitions, Q() returns the number of unequal
partitions, and R() returns the number of
restricted partitions. Function S() returns the number of
block partitions.
Usage
P(n, give = FALSE)
Q(n, give = FALSE)
R(m, n, include.zero = FALSE)
S(f, n = NULL, include.fewer = FALSE)
Arguments
n
Integer whose partition number is desired. In function
S(), the default of NULL means to return the number of
partitions of any size
m
In function R(), the order of the
decomposition
give
Boolean, with default FALSE meaning to return just
P(n) or Q(n) and TRUE meaning to return
P(1:n) or Q(1:n) (this option takes no extra
computation)
include.zero
In restrictedparts(), Boolean with
default FALSE meaning to count only partitions of n
into exactlym parts; and TRUE meaning to
include partitions of n into at mostm parts
(because parts of zero are included)
include.fewer
In function blockparts(), Boolean
with default FALSE meaning to return partitions into
exactlyn and TRUE meaning to return partitions
into at mostn
f
In function S(), the stack vector
Details
Functions P() and Q() use Euler's
recursion formula. Function R() enumerates the partitions
using Hindenburg's method (see Andrews) and counts them until the
recursion bottoms out.
Function S() finds the coefficient of x^n in the
generating function prod_{i=1}^{L}(1+x+x^2+...+x^(f[i])), where L is the
length of f, using the polynom package.
All these functions return a double.
Note
Functions P() and Q() use unsigned long long
integers, a type which is system-dependent. For me, P() works
for n equal to or less than 416, and Q() works for
n less than or equal to 792. YMMV; none of the methods test
for overflow, so use with care!
Author(s)
Robin K. S. Hankin; S() is due to an anonymous JSS referee