Let S be a set of some positive integers.
We'll call S autonomous if the number of elements in S is itself an element of S. e.g. the set {2,3,5} is autonomous, as is the set {2,7}, but the sets {1, 4} or {2,4,5} are not.

Determine a general formula for the number
of autonomous subsets of {1, 2, 3, ... , n}.

I think Charlie found the number of NON-autonomous subsets.  The correct formula for the number of autonomous subsets is 2^(n-1).

f(1) = 1 {1}

f(2) = 2 {1}, {1,2}

f(3) = 4 {1}, {1,2}, {2,3}, {1,2,3}

f(4) = 8 {1}, {1,2}, {2,3}, {2,4}, {1,2,3}, {1,3,4}, {2,3,4}, {1,2,3,4}

etc.

 

 

 

Posted by tomarken on 2014-07-30 08:48:04