OGF Problem Examples

Combinatorics · Axiom Academy

Step-by-step solutions to classic generating function problems Problem: Counting Binary Strings Find the ordinary generating function for the number of binary strings of length that do not contain two consecutive 1's. Excellent work! You've mastered OGF problem-solving. Here's what we learned: Recurrence Relations: Many counting problems can be solved by finding a recurrence relation and then deriving the OGF from it. Initial Conditions: Always carefully identify and incorporate initial conditions when setting up the generating function equation. Algebraic Manipulation: The key skill is manipulating the series equation to isolate A(x) and express it in closed form. Product Rule: When counting selections with restrictions, multiply the generating functions for independent choices. Coefficient Extraction: Remember that [x n ]A(x) gives the coefficient, which represents the count for size n. These techniques form the foundation for solving complex combinatorial problems. Practice identifying when to use recurrences versus direct product constructions!

This is the written version of the interactive lesson above. See the full Combinatorics course.