• combinatorial reciprocity
• generalized permutahedra
• Hopf monoids
• vector species
• zonotopes

Combinatorial reciprocity is a principle linking numerical and algebraic properties of combinatorial objects. A classical example is the chromatic polynomial in graph theory, which counts proper graph colorings and, when evaluated at negative integers, reveals information about acyclic orientations. Following the paper ‘Hopf Monoids and Generalized Permutahedra’ by M. Aguiar and F. Ardila, this thesis investigates combinatorial reciprocity in the framework of Hopf monoids in vector species, a generalization of Hopf algebras that provides a powerful language for studying combinatorial and geometric structures. This theoretical framework not only generalizes classical reciprocity theorems but also offers new perspectives on polyhedral structures and their algebraic properties.

The thesis is structured into four chapters. The first introduces set species, a categorial approach associating combinatorial structures with finite sets, culminating in bimonoidal set species as the foundation for later developments. The second chapter explores the interplay between combinatorics and geometry, presenting classical geometric structures like zonotopes and generalized permutahedra, which are key in understanding Hopf monoids. The third chapter introduces vector species, allowing for a more refined algebraic treatment of combinatorial reciprocity, particularly through the study of antipodes in Hopf monoids. The final chapter, analyzing multiplicative characters, applies these tools to combinatorial reciprocity theorems.