# combn: 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

 `1` ```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.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17``` ```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), 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))) ```