If by 'partition' you mean an unordered way to write a number n as a sum of positive integers, the first few partitions are:

P(1) = 1, as

1 can only be written as 1.

P(2) = 2, as

2 can be written as 2 or 1+1.

P(3) = 3, as

3 can be written as 3 or 2+1 or 1+1+1.

P(4) = 5, as

4 can be written as 4 or 3+1 or 2+2, 2+1+1, 1+1+1+1.

which don't match your problem description.

Please clarify your problem.