Partitions?

Since $15=10+5=1+2+3+4+5$

We need to distribute the $5$ to the numbers in different ways to make different ways to represent the number.

So $15=1+2+3+(4+5)=1+2+3+9$ $15=1+2+(3+5)+4=1+2+8+4$ $15=1+(2+5)+3+4=1+7+3+4$ $15=(1+5)+2+3+4=6+2+3+4$

Now we need to split the 5 between the numbers.The only ways to split 5 such that there are 4 or less numbers in the partition is $5=4+1=3+2=3+1+1=2+2+1$ .The only ways to split it such that no number equals the other in the partition (and also that it's not equivalent to another partition) is $1+2+5+7$

So the total number is $\boxed{6}$

There are "cleverer" ways to solve this, but there are so few solutions that it's easiest to just list them!

1+2+3+9

1+2+4+8

1+2+5+7

1+3+4+7

1+3+5+6

2+3+4+6 [edited for typo]

{{9, 3, 2, 1}, {8, 4, 2, 1}, {7, 5, 2, 1}, {7, 4, 3, 1}, {6, 5, 3, 1}, {6, 4, 3, 2}}