Todd Rowland
Wolfram Research
Maryland
Registered: Oct 2003
Posts: 116 
As you seem to have guessed, this is a known problem. The numbers you give are the binomial distribution ( Binomial[14,n]), the number of ways of picking n objects from 14 (order of choices doesn't matter). The problem of going from one to the next has at least one answer. In Mathematica, it is implemented as NextKSubset in the Combinatorica package.
One place to read about it from a dynamical NKS perspective is the recent article "Permutation Numbers", Vincenzo De Florio, Complex Systems, Vol 152. Basically, it is an iterated Turing Machine. Of course, the TM is very simple. It is a constructed one, so one wonders what else is out there, undiscovered.
Finding a CA to match a fixed pattern is an interesting problem. When looking for a literal match, it is relatively straightforward to check if the evolution is a CA. In this case, the sequence is not a CA, because it is not local.
But if one is looking for overall features, or some other purpose where an exact match is not needed, it is not as straightforward. The place to start though is to perform a search of the appropriate rule space and look at what happens.
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