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Advanced Combinatorics & Olympiad Prep
·
Module 7
Recurrence Relations and Generating Functions
1
Recurrence Relations for Combinatorial Problems
Formulate recurrence relations for combinatorial problems, including the plane division problem.
2
Solving Linear Homogeneous Recurrence Relations: Characteristic Equation Method
Solve linear homogeneous recurrence relations using the characteristic equation method.
3
Building OGFs for Integer Sequences
Construct ordinary generating functions (OGFs) to represent integer sequences.
4
Solving Recurrences with OGFs
Use OGFs and partial fraction decomposition to find closed-form solutions for linear recurrences.
Catalan Numbers: From Combinatorics to Generating Functions
Verify that a combinatorial problem maps to the Catalan numbers and derive their generating function.
6
Enumerating Labeled Structures with Exponential Generating Functions
Apply exponential generating functions to enumerate permutations and other labeled structures.
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