Solving Generating Function Problems - How to Get the Answer

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Discussion Overview

The discussion revolves around generating functions, their applications, and how to utilize them to solve combinatorial problems. Participants explore the concept of generating functions, specifically in the context of coloring a chessboard with certain constraints, and the transition from ordinary to exponential generating functions.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Homework-related

Main Points Raised

  • One participant seeks clarification on the application of generating functions after obtaining them, specifically regarding a problem involving coloring a chessboard with constraints on colors.
  • Another participant suggests that the generating function is the reciprocal of a specific polynomial and explains that the coefficients in the power series expansion correspond to the values sought.
  • A participant corrects their earlier statement about the generating function and introduces the idea that exponential generating functions may be necessary for their problem.
  • One participant describes generating functions as a useful representation of solutions to combinatorial problems, emphasizing their importance for those studying combinatorics.
  • Another participant requests examples of generating functions in use, indicating a desire for practical applications.
  • A participant provides a link to a Google search for examples of generating functions and mentions that the binomial expansion is a generating function for combinations.
  • A recommendation for a book on generating functions is shared, suggesting it as a resource for further learning.

Areas of Agreement / Disagreement

Participants express varying levels of understanding and application of generating functions, with some uncertainty about the specific types needed for different problems. There is no consensus on the best approach to the initial problem presented.

Contextual Notes

Participants mention the need for different types of generating functions (ordinary vs. exponential) without resolving the implications of this distinction for the problem at hand. There are also references to specific mathematical expressions and their interpretations that may depend on context.

Who May Find This Useful

This discussion may be useful for students and educators in combinatorics, mathematics, or related fields who are exploring generating functions and their applications in problem-solving.

mansi
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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.
 
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Do you mean that the generating function is
[tex]\frac{1}{f(x)}= \frac{1}{(1-x)^2(1-x^3)}[/tex]?

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?
 
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?
 
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!
 
That's precisely what my professor said in class!Can you give me examples where they're used?
 
Thanks,Sir...that was a big help :-)
 
Generatingfunctionology

If your professor isn't already using it, I recommend Wilf's wonderful little book http://www.math.upenn.edu/~wilf/DownldGF.html" .
 
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