Number of Partitions/Compositions

Calculate the number of partitions/compositions of a vector chosen \(m\) at a time with or without replacement. Additionally, these functions can calculate the number of partitions of multisets.

partitionsCount(v, m = NULL, ...)
compositionsCount(v, m = NULL, ...)

# S3 method for default
partitionsCount(v, m = NULL, repetition = FALSE,
                freqs = NULL, target = NULL, ...)
# S3 method for default
compositionsCount(v, m = NULL, repetition = FALSE,
                  freqs = NULL, target = NULL, weak = FALSE, ...)

# S3 method for table
partitionsCount(v, m = NULL, target = NULL, ...)
# S3 method for table
compositionsCount(v, m = NULL, target = NULL, weak = FALSE, ...)

Arguments

v

Source vector. If v is a positive integer, it will be converted to the sequence 1:v. If v is a negative integer, it will be converted to the sequence v:-1. Only integer and numeric vectors are accepted.

m

Width of the partition. If m = NULL, the length will be determined by the partitioning case (e.g. When we are generating distinct partitions of \(n\), the width will be equal to the smallest \(m\) such that sum(1:m) >= n).

...

Further arguments passed to methods.

repetition

Logical value indicating whether partitions/compositions should be with or without repetition. The default is FALSE.

freqs

A vector of frequencies used for producing all partitions of a multiset of v. Each element of freqs represents how many times each element of the source vector, v, is repeated. It is analogous to the times argument in rep. The default value is NULL.

target

Number to be partitioned. If NULL, max(v) will be used.

weak

(Compositions only) Logical flag indicating whether to allow terms of the sequence to be zero.

Value

A numerical value representing the total number of partitions/compositions.

Note

When the number of results exceeds \(2^{53} - 1\), a number of class bigz is returned.

See also

partitionsGeneral, compositionsGeneral

Examples

## Same interface as partitionsGeneral
partitionsCount(25, 5)
#> [1] 30
compositionsCount(25, 5, TRUE)
#> [1] 10626
partitionsCount(15, 7, TRUE)
#> [1] 21
partitionsCount(25, 5, freqs = rep(2, 25))
#> [1] 148

## Return object of class 'bigz'
partitionsCount(2500, 15, TRUE)
#> Big Integer ('bigz') :
#> [1] 4188807783906550542066742
compositionsCount(2500, 15, TRUE)
#> Big Integer ('bigz') :
#> [1] 4097094997743241086270928785678827376