Partitions generator

This function partitions an non-negative interger n into k parts or parts of any sizes. The results are in lexicographical or reversed lexicographical order.

partitions(n, k = NULL, distinct = FALSE, descending = FALSE,
  layout = NULL, nitem = -1L, skip = NULL, index = NULL,
  nsample = NULL, drop = NULL)

Arguments

n	an non-negative integer to be partitioned
k	number of parts
distinct	an logical to restrict distinct values
descending	an logical to use reversed lexicographical order
layout	if "row", "column" or "list" is specified, the returned value would be a "row-major" matrix, a "column-major" matrix or a list respectively
nitem	number of partitions required, usually used with skip
skip	the number of partitions skipped
index	a vector of indices of the desired partitions
nsample	sampling random partitions
drop	vectorize a matrix or unlist a list

See also

ipartitions for iterating partitions and npartitions to calculate number of partitions

Examples

# all partitions of 6
partitions(6)
#>       [,1] [,2] [,3] [,4] [,5] [,6]
#>  [1,]    1    1    1    1    1    1
#>  [2,]    1    1    1    1    2    0
#>  [3,]    1    1    1    3    0    0
#>  [4,]    1    1    2    2    0    0
#>  [5,]    1    1    4    0    0    0
#>  [6,]    1    2    3    0    0    0
#>  [7,]    1    5    0    0    0    0
#>  [8,]    2    2    2    0    0    0
#>  [9,]    2    4    0    0    0    0
#> [10,]    3    3    0    0    0    0
#> [11,]    6    0    0    0    0    0
# reversed lexicographical order
partitions(6, descending = TRUE)
#>       [,1] [,2] [,3] [,4] [,5] [,6]
#>  [1,]    6    0    0    0    0    0
#>  [2,]    5    1    0    0    0    0
#>  [3,]    4    2    0    0    0    0
#>  [4,]    4    1    1    0    0    0
#>  [5,]    3    3    0    0    0    0
#>  [6,]    3    2    1    0    0    0
#>  [7,]    3    1    1    1    0    0
#>  [8,]    2    2    2    0    0    0
#>  [9,]    2    2    1    1    0    0
#> [10,]    2    1    1    1    1    0
#> [11,]    1    1    1    1    1    1

# fixed number of parts
partitions(10, 5)
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,]    1    1    1    1    6
#> [2,]    1    1    1    2    5
#> [3,]    1    1    1    3    4
#> [4,]    1    1    2    2    4
#> [5,]    1    1    2    3    3
#> [6,]    1    2    2    2    3
#> [7,]    2    2    2    2    2
# reversed lexicographical order
partitions(10, 5, descending = TRUE)
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,]    6    1    1    1    1
#> [2,]    5    2    1    1    1
#> [3,]    4    3    1    1    1
#> [4,]    4    2    2    1    1
#> [5,]    3    3    2    1    1
#> [6,]    3    2    2    2    1
#> [7,]    2    2    2    2    2

# column major
partitions(6, layout = "column")
#>      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
#> [1,]    1    1    1    1    1    1    1    2    2     3     6
#> [2,]    1    1    1    1    1    2    5    2    4     3     0
#> [3,]    1    1    1    2    4    3    0    2    0     0     0
#> [4,]    1    1    3    2    0    0    0    0    0     0     0
#> [5,]    1    2    0    0    0    0    0    0    0     0     0
#> [6,]    1    0    0    0    0    0    0    0    0     0     0
partitions(6, 3, layout = "column")
#>      [,1] [,2] [,3]
#> [1,]    1    1    2
#> [2,]    1    2    2
#> [3,]    4    3    2

# list output
partitions(6, layout = "list")
#> [[1]]
#> [1] 1 1 1 1 1 1
#> 
#> [[2]]
#> [1] 1 1 1 1 2
#> 
#> [[3]]
#> [1] 1 1 1 3
#> 
#> [[4]]
#> [1] 1 1 2 2
#> 
#> [[5]]
#> [1] 1 1 4
#> 
#> [[6]]
#> [1] 1 2 3
#> 
#> [[7]]
#> [1] 1 5
#> 
#> [[8]]
#> [1] 2 2 2
#> 
#> [[9]]
#> [1] 2 4
#> 
#> [[10]]
#> [1] 3 3
#> 
#> [[11]]
#> [1] 6
#> 
partitions(6, 3, layout = "list")
#> [[1]]
#> [1] 1 1 4
#> 
#> [[2]]
#> [1] 1 2 3
#> 
#> [[3]]
#> [1] 2 2 2
#> 

# zero sized partitions
dim(partitions(0))
#> [1] 1 0
dim(partitions(5, 0))
#> [1] 0 0
dim(partitions(5, 6))
#> [1] 0 6
dim(partitions(0, 0))
#> [1] 1 0
dim(partitions(0, 1))
#> [1] 0 1