Sofia Eriksson: Den tolvfaldiga vägen

Abstract: The purpose of this thesis is to provide an account of the Twelvefold Way, a combinatorial framework that classifies twelve closely related distribution problems, structured by the distinctions between objects (balls) and containers (boxes) under various conditions. The work follows a pedagogical and combinatorial progression, starting from elementary counting principles such as the multiplication principle, permutations, and unordered selections via stars and bars. The study then delves into set partitions, Stirling numbers of the second kind, and their connection to Bell numbers and surjective functions. Thereafter, the focus shifts toward analytic combinatorics, utilizing ordinary generating functions to encode numerical sequences into functions for clearer pattern interpretation. Finally, integer partitions are examined alongside their associated bijective transformations and Ferrers diagrams. Through this structured progression, the thesis demonstrates how unified definitions bridge classic counting methods with analytic combinatorics.