perplexus.info :: Just Math : Easy count

Easy count (Posted on 2014-07-29)

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}.

See The Solution Submitted by Ady TZIDON

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computer assisted solution

| Comment 1 of 5

Let f(n) be the sought function of n.

f(1) = 0

f(2) = 1 {2}

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

f(4) = 7 {2}{3}{4}{1,3}{1,4}{3,4}{1,2,4}

f(n) = Sigma{i=1 to n-1} C(n-1,i)

10 for N=2 to 12

15 T=0

20 for I=1 to N-1

30 T=T+combi(N-1,I)

40 next

50 print N,T

60 next

finds

f(n) = 2^(n-1) - 1

Posted by Charlie on 2014-07-29 14:38:40

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