On Mon, Dec 15, 2014 at 10:06 AM, David L Carlson <dcarl...@tamu.edu> wrote:
> Actually there are not so many matrices as you suggest.
>
> > comb <- combn(28, 4)
> > dim(comb)
> [1] 4 20475
> > sum(comb[1,]==1)
> [1] 2925
> > comb[, 1]
> [1] 1 2 3 4
>
> There are 20,475 combinations, but you cannot choose any four to make a
> 4x7 matrix since each value can be used only once. The combn() function
> returns the combinations sorted, so we can get the number of combinations
> that contain 1 with sum(comb[1,]==1) and that is 2,925. The set of 4x7
> matrices cannot use the same combination more than once, so 2,925 is the
> maximum possible number of matrices and there may be fewer. As a first
> approach to finding them, you could take the first combination comb[, 1]
> which is 1, 2, 3, 4. Now add a second combination that does not include 1:4
> and then a third combination that does not include any in the first two
> combinations and finally a fourth that does not include any in the first
> three combinations. Actually this is easy since we will just take 1:4, 5:8,
> 9:12, 13:16, 17:20, 21:24, 24:18.
>
> > cols <- sapply(c(1, 5, 9, 13, 17, 21, 24), function(x)
> + head(which(comb[1,]==x), 1))
> > cols
> [1] 1 9850 15631 18656 19981 20406 20471
> > comb[,cols]
> [,1] [,2] [,3] [,4] [,5] [,6] [,7]
> [1,] 1 5 9 13 17 21 24
> [2,] 2 6 10 14 18 22 25
> [3,] 3 7 11 15 19 23 26
> [4,] 4 8 12 16 20 24 27
>
> But now it gets more complicated. While building the second matrix, we
> have to make sure that it does not use any combinations that have already
> been used. Combinations used on earlier matrices may be necessary to
> complete later matrices and that is why the number of sets may be less than
> 2,925. This sequential approach would guarantee to obtain matrices meeting
> the OP's criteria, but would not necessarily produce the maximum number of
> matrices possible.
>
Thanks. I was thinking that I was not on the right road, but just didn't
see what I was doing wrong. I appreciate the tutoring. And I'm bowing out
now because this is getting far beyond my expertise both in math and R.
>
> -------------------------------------
> David L Carlson
> Department of Anthropology
> Texas A&M University
> College Station, TX 77840-4352
>
>
--
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