# The number of ways to express a specific summation

1. Sep 1, 2010

### eehsun

Hello all,

I have been thinking about a particular mathematical question (that I've made up) and I haven't been able to reach a solution yet..

I want to find the rule for the function F(x,y) which gives the number of different "ways" that the integer x can be expressed as the summation of "y" pieces of integers (these integers have to be bigger than or equal to 1) (sorry for my awful technical English

Let me clarify it with an example:

When we consider F(9,4), it can be observed that

9 = 1+1+1+6
9 = 1+1+2+5
9 = 1+1+3+4
9 = 1+2+2+4
9 = 1+2+3+3
9 = 2+2+2+3

Following from here, since there are 6 different ways of expressing this summation,
F(9,4)=6

In the above example, (1+1+1+6) and, for example, (1+6+1+1) are considered to be the same and thus are counted only once.

NOTE: When we consider (1+1+1+6) and (1+6+1+1) to be different ways of summation, for instance, the problem becomes very easy and can be solved by pigeon hole principle. But the tricky part for me is to find a formula which considers the two expressions above and such to amount to the same.

Thanks!

2. Sep 1, 2010