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Introduction to Combinatorics
Abstract
Combinatorics, a branch of mathematics that studies the possibilities of selecting, arranging, and grouping elements of a set, has ancient roots in various cultures. The chapter traces the history of combinatorics, from its origins in the ancient East to modern developments. Key figures such as Pascal, Leibniz and Fibonacci are highlighted as important contributions. The chapter illustrates the connection of combinatorics with graph theory and other complex concepts thanks to figures like Paul Erdős and Ronald Graham. The most recent developments in combinatorics are examined, focusing on the impact of artificial intelligence and machine learning. It highlighted how these technologies open new avenues for optimizing graph problems. Combinatorics is fundamental for game theory, circuit design, social networks, and drug discovery. The chapter introduces theorems such as those of Szemerédi and Green-Tao that have brought new points of view. Finally, it presents several simulations to clarify better concepts, such as the drunken path and the Sierpinsky triangle.
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