# Generating Functions!

1. Jan 16, 2006

### mansi

We are about to begin this topic soon in class...
I'd like to know all about generating functions and their application. I tried reading it up...One thing I'd like to know is, once you have a generating function...then what? You get some information...but what is it?
I tried to find no. of ways of colouring a 1 by n chessboard in 3 colours say red, blue & green if an even no. of squares must be red,even no. of squares must be blue & odd no. of squares must be green...

I think the generating function turns out to be (f(x))^-1 where f(x)= ((1-x^2)^2)*(1-x^3)). So now, what do I do to get the Required answer and why??
Thanks for the help.

2. Jan 16, 2006

### HallsofIvy

Staff Emeritus
Do you mean that the generating function is
$$\frac{1}{f(x)}= \frac{1}{(1-x)^2(1-x^3)}$$?

The whole point of a "generating function" is that the coeffient of xn in its power series expansion is nth value you are looking for. Do you know how to find the power series for this function?

3. Jan 17, 2006

### mansi

I meant f(x) is that expression and not (f(x))^-1.
But well...I just got to know that the problem can be solved not using ordinary generating functions but will require "something" called exponential generating functions.
Can Someone here please give me a motivation for the concept of generating functions?

4. Jan 17, 2006

### Hurkyl

Staff Emeritus
A generating function is just another way to represent the solutions to a problem.

This representation has proven that it can often be useful, therefore it becomes something every budding combinatorist should learn!

5. Jan 17, 2006

### mansi

That's precisely what my professor said in class!Can you give me examples where they're used?

6. Jan 17, 2006

### matt grime

7. Jan 17, 2006

### mansi

Thanks,Sir...that was a big help :-)

8. Jan 19, 2006