I think the problem statement is not that clear. What is intended is that you find the number of distinct sets of notes for which the following is true:

- The notes in the set sum up to N
- For every value v from 1 to N, there is exactly one combination of note values that sums up to v.

With n = 5:

{1,1,1,1,1}:

1 => use 1

2 => use 1, 1

3 => use 1, 1, 1

4 => use 1, 1, 1, 1

5 => use 1, 1, 1, 1, 1

Other combinations to create 1-5 are not possible.

{1,2,2):

1 => use 1

2 => use 2

3 => use 1, 2

4 => use 2, 2

5 => use 1, 2, 2

Other combinations to create 1-5 are not possible.

{1,1,3}:

1 => use 1

2 => use 1, 1

3 => use 3

4 => use 1, 3

5 => use 1, 1, 3

{4,1} is not correct, because you cannot create 2 and 3

{3,2} is not correct, because you cannot create 1 and 4

For n = 10:

{1, 1, 1, 1, 1, 1, 1, 1, 1, 1} is a correct set (for all N, the set of N times 1 will be correct)

Apparently, there is no other set of values that satisfies the two conditions.

Without some further thought, I do not have a simple way to explain why there is no other set. My guess is that if you explore the other candidate sets and find why they are incorrect, you may discover how to solve this problem.

Good luck.