On Sat, 21 Jun 2008, Gavin Simpson 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.
Right. And in the example below there are 32 distinct patterns.Which arises from sum( choose( 5, 0:5 ) ) different placements of 0:5 split positions.
You can represent the splits as a binary number with n-1 bits: 00000 implies no splits, 10000 implies a split between 1 and 2, 10100 implies splits between 1 and 2 and between 3 and 4, et cetera.
So, 32 arises as 2^5, too. Something like this:
base10 <- seq(0, length=2^(n-1) ) base2.bits <- outer(0:(n-2), base10, function(y,x) ( x %/% (2^y)) %%2 ) sapply(apply( base2.bits==1, 2, which ), function(x) rep(1:(1+length(x)), diff(c(0,x,n))))
Getting this in the same column order as your example is left as an exercise for the reader.
HTH, Chuck
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?
Thanks in advance,
Gavin
--
%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%
Dr. Gavin Simpson [t] +44 (0)20 7679 0522
ECRC, UCL Geography, [f] +44 (0)20 7679 0565
Pearson Building, [e] gavin.simpsonATNOSPAMucl.ac.uk
Gower Street, London [w] http://www.ucl.ac.uk/~ucfagls/
UK. WC1E 6BT. [w] http://www.freshwaters.org.uk
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Charles C. Berry (858) 534-2098
Dept of Family/Preventive Medicine
E mailto:[EMAIL PROTECTED] UC San Diego
http://famprevmed.ucsd.edu/faculty/cberry/ La Jolla, San Diego 92093-0901
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