# [R] Generating groupings of ordered observations

Gabor Grothendieck ggrothendieck at gmail.com
Sat Jun 21 23:56:22 CEST 2008

```On Sat, Jun 21, 2008 at 12:40 PM, Gavin Simpson <gavin.simpson at ucl.ac.uk> wrote:
> Dear List,
>
> I have a problem I'm finding it difficult to make headway with.
>
> Say I have 6 ordered observations, and I want to find all combinations
> of splitting these 6 ordered observations in g groups, where g = 1, ...,
> 6. Groups can only be formed by adjacent observations, so observations 1
> and 4 can't be in a group on their own, only if 1,2,3&4 are all in the
> group.
>
> For example, with 6 observations, the columns of the matrices below
> represent the groups that can be formed by placing the 6 ordered
> observations into 2-5 groups. Think of the columns of these matrices as
> being an indicator of group membership. We then cbind these matrices
> with the trivial partitions into 1 and 6 groups:
>
> mat2g <- matrix(c(1,1,1,1,1,
>                  2,1,1,1,1,
>                  2,2,1,1,1,
>                  2,2,2,1,1,
>                  2,2,2,2,1,
>                  2,2,2,2,2),
>                nrow = 6, ncol = 5, byrow = TRUE)
>
> mat3g <- matrix(c(1,1,1,1,1,1,1,1,1,1,
>                  2,2,2,2,1,1,1,1,1,1,
>                  3,2,2,2,2,2,2,1,1,1,
>                  3,3,2,2,3,2,2,2,2,1,
>                  3,3,3,2,3,3,2,3,2,2,
>                  3,3,3,3,3,3,3,3,3,3),
>                nrow = 6, ncol = 10, byrow = TRUE)
>
> mat4g <- matrix(c(1,1,1,1,1,1,1,1,1,1,
>                  2,2,2,2,2,2,1,1,1,1,
>                  3,3,3,2,2,2,2,2,2,1,
>                  4,3,3,3,3,2,3,3,2,2,
>                  4,4,3,4,3,3,4,3,3,3,
>                  4,4,4,4,4,4,4,4,4,4),
>                nrow = 6, ncol = 10, byrow = TRUE)
>
> mat5g <- matrix(c(1,1,1,1,1,
>                  2,2,2,2,1,
>                  3,3,3,2,2,
>                  4,4,3,3,3,
>                  5,4,4,4,4,
>                  5,5,5,5,5),
>                nrow = 6, ncol = 5, byrow = TRUE)
>
> cbind(rep(1,6), mat2g, mat3g, mat4g, mat5g, 1:6)
>
> I'd like to be able to do this automagically, for any (reasonable,
> small, say n = 10-20) number of observations, n, and for g = 1, ..., n
> groups.
>
> I can't see the pattern here or a way forward. Can anyone suggest an
> approach?
>

Peter Wolf has APL-style encode/decode functions on his web site that
can readily be used for this.  The output of the encode below are the binary
digits expansions of the numbers 0:15, one per column, and the remainder
transforms that matrix to the required one (but columns are in a different
order than yours):

> source("http://www.wiwi.uni-bielefeld.de/~wolf/software/R-wtools/decodeencode/decodeencode.R")
> n <- 6
> n1 <- n-1
> apply(rbind(1, encode(0:(2^n1-1), rep(2, n1))), 2, cumsum)
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12]
[,13] [,14] [,15] [,16] [,17] [,18] [,19] [,20] [,21] [,22] [,23]
[,24] [,25] [,26]
[1,]    1    1    1    1    1    1    1    1    1     1     1     1
1     1     1     1     1     1     1     1     1     1     1     1
1     1
[2,]    1    1    1    1    1    1    1    1    1     1     1     1
1     1     1     1     2     2     2     2     2     2     2     2
2     2
[3,]    1    1    1    1    1    1    1    1    2     2     2     2
2     2     2     2     2     2     2     2     2     2     2     2
3     3
[4,]    1    1    1    1    2    2    2    2    2     2     2     2
3     3     3     3     2     2     2     2     3     3     3     3
3     3
[5,]    1    1    2    2    2    2    3    3    2     2     3     3
3     3     4     4     2     2     3     3     3     3     4     4
3     3
[6,]    1    2    2    3    2    3    3    4    2     3     3     4
3     4     4     5     2     3     3     4     3     4     4     5
3     4
[,27] [,28] [,29] [,30] [,31] [,32]
[1,]     1     1     1     1     1     1
[2,]     2     2     2     2     2     2
[3,]     3     3     3     3     3     3
[4,]     3     3     4     4     4     4
[5,]     4     4     4     4     5     5
[6,]     4     5     4     5     5     6

```