Here n is a positive integer and p(n) is the number of unrestricted partitions. Can one always find a subset, s, of {1,2,...n} such that the number of partitions of n with parts from s is p(n)/2 if p(n) is even and is (p(n)+1)/2 or (p(n)-1)/2 if n is odd?

For example: if n=7 we may choose s to be {1,3,4,5,7} The partitions of 7 which are to be counted are

7

5+1+1

4+3

3+3+1

4+1+1+1

3+1+1+1+1

1+1+1+1+1+1+1

and (p(n)-1)/2= (15-1)/2-7 so we can do what was asked of us.