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.
P(n, give = FALSE)
Q(n, give = FALSE)
R(m, n, include.zero = FALSE)
S(f, n = NULL, include.fewer = FALSE)
Integer whose partition number is desired. In function
S()
, the default of NULL
means to return the number of
partitions of any size
In function R()
, the order of the
decomposition
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)
In restrictedparts()
, Boolean with
default FALSE
meaning to count only partitions of \(n\)
into exactly \(m\) parts; and TRUE
meaning to
include partitions of \(n\) into at most \(m\) parts
(because parts of zero are included)
In function blockparts()
, Boolean
with default FALSE
meaning to return partitions into
exactly n
and TRUE
meaning to return partitions
into at most n
In function S()
, the stack vector
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 \(\prod_{i=1}^L\sum_{j=0}^{f_i}
x^j\), where \(L\) is the
length of f
, using the polynom package.
All these functions return a double.
# NOT RUN {
P(10,give=TRUE)
Q(10,give=TRUE)
R(10,20,include.zero=FALSE)
R(10,20,include.zero=TRUE)
S(1:4,5)
# }
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