# combn: Generate All Combinations of n Elements, Taken m at a Time

 combn R Documentation

## Generate All Combinations of n Elements, Taken m at a Time

### Description

Generate all combinations of the elements of `x` taken `m` at a time. If `x` is a positive integer, returns all combinations of the elements of `seq(x)` taken `m` at a time. If argument `FUN` is not `NULL`, applies a function given by the argument to each point. If simplify is FALSE, returns a list; otherwise returns an `array`, typically a `matrix`. `...` are passed unchanged to the `FUN` function, if specified.

### Usage

```combn(x, m, FUN = NULL, simplify = TRUE, ...)
```

### Arguments

 `x` vector source for combinations, or integer `n` for `x <- seq_len(n)`. `m` number of elements to choose. `FUN` function to be applied to each combination; default `NULL` means the identity, i.e., to return the combination (vector of length `m`). `simplify` logical indicating if the result should be simplified to an `array` (typically a `matrix`); if FALSE, the function returns a `list`. Note that when `simplify = TRUE` as by default, the dimension of the result is simply determined from `FUN(1st combination)` (for efficiency reasons). This will badly fail if `FUN(u)` is not of constant length. `...` optionally, further arguments to `FUN`.

### Details

Factors `x` are accepted.

### Value

A `list` or `array`, see the `simplify` argument above. In the latter case, the identity `dim(combn(n, m)) == c(m, choose(n, m))` holds.

### Author(s)

Scott Chasalow wrote the original in 1994 for S; R package combinat and documentation by Vince Carey stvjc@channing.harvard.edu; small changes by the R core team, notably to return an array in all cases of `simplify = TRUE`, e.g., for `combn(5,5)`.

### References

Nijenhuis, A. and Wilf, H.S. (1978) Combinatorial Algorithms for Computers and Calculators; Academic Press, NY.

`choose` for fast computation of the number of combinations. `expand.grid` for creating a data frame from all combinations of factors or vectors.

### Examples

```combn(letters[1:4], 2)
(m <- combn(10, 5, min))   # minimum value in each combination
mm <- combn(15, 6, function(x) matrix(x, 2, 3))
stopifnot(round(choose(10, 5)) == length(m), is.array(m), # 1-dimensional
c(2,3, round(choose(15, 6))) == dim(mm))

## Different way of encoding points:
combn(c(1,1,1,1,2,2,2,3,3,4), 3, tabulate, nbins = 4)

## Compute support points and (scaled) probabilities for a
## Multivariate-Hypergeometric(n = 3, N = c(4,3,2,1)) p.f.:
# table.mat(t(combn(c(1,1,1,1,2,2,2,3,3,4), 3, tabulate, nbins = 4)))

## Assuring the identity
for(n in 1:7)
for(m in 0:n) stopifnot(is.array(cc <- combn(n, m)),
dim(cc) == c(m, choose(n, m)),
identical(cc, combn(n, m, identity)) || m == 1)
```