How many ways a number can be written as components sum?

[Adel Makram]

If we have a positive integer, how many ways can this number be written as a sum of its components? By components, I mean all numbers less than that number.
For example, 5 has 6 ways to be written;
5x1, 3x1+2, 2x2+1, 2x1+3,1+4 and 2+3. In digits form; [11111, 1112, 221,113, 14, 23]
So there are 6 ways to write 5.
For number 6; [111111, 11112,1113,114,15, 2+4, 3+3, 1+2+3, 2+2+2 and 1+2+2+1]. which is 10.
In general how many ways to write a number as a sum of all possible combination of numbers less than that number?

 

[jedishrfu]

This kind of computation, I think is covered by partition theory:

https://en.wikipedia.org/wiki/Partition_(number_theory)

 

[LCKurtz]

This kind of computation is covered by partition theory:

https://en.wikipedia.org/wiki/Partition_(number_theory)

Does partition theory include sums of products?

 

[jedishrfu]

Does partition theory include sums of products?

I don't know if it covers sums of products. I was thinking only of the number of ways that a given number can be represented via addition by lower numbers.