the number of ways to express a specific summation

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!

adriank

The term you are looking for is "partition". So there's a starting point for you.

eehsun

Thank you very much adriank !