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Exploring Generating Functions in Combinatorial Problems

Learn about generating functions, including definition and applications in solving recurrences and combinatorial counting. Example problems and solutions provided. Reading assignments for further practice included.

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Exploring Generating Functions in Combinatorial Problems

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  1. Chapter 7 Generating Function and Applications

  2. 7.2 Generating Function

  3. Definition of Generating Function G(x): e.g. vs.

  4. e.g. 1.2

  5. e.g. 1.4 =?

  6. e.g. 2.2 (NCKU) Show that Constant term is

  7. Considering constant term is

  8. 7.3 Application(1): Solving Recurrence

  9. Example 1 Solve with and . From and , we have From , we have

  10. From original recurrence and boundary conditions, we further have

  11. We have Final solution is

  12. Reading Assignment: Solve nonhomogeneous recurrence By using generating function.

  13. 7.4 Application(2): Combinatorial Counting

  14. Example 4 Given n nodes, how many rooted binary trees that we can build up? When n=5, there are five trees.

  15. Recurrence: We still need the boundary conditions: , , and

  16. We have and It yields to

  17. Derivation:

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