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From Ordinary Generating Functions to Combinatorial Calculus
Abstract
The chapter explores fundamental concepts of combinatorics, focusing on ordinary generating functions (FGOs) and their applications in counting problems. FGOs are powerful tools for analyzing sequences by transforming them into algebraic expressions. They help in solving problems related to permutations, combinations, and arrangements. The chapter demonstrates how FGOs can generate sequences like Fibonacci numbers and solve real-world problems like binary sequence formation. The chapter then discusses different types of permutations, including simple permutations, circular permutations, and permutations with repetition. It also covers arrangements, which are ordered selections of elements and their variations with repetition. Combinations, where order does not matter, are analyzed, particularly in games like poker and probability calculations. Special cases include combinations with repetition and dismutations, a concept where no element remains in its original position.
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