Emanuele Verri

• This thesis develops new connections between categorical factorization systems and the structure of combinatorial Hopf algebras. Focusing on the category DiGraph of binary relations—and several important subcategories such as simple graphs and posets—the work shows that the existence of RegEpi–Mono and Epi–RegMono factorizations leads to natural translation maps between counting profiles associated with graph homomorphisms, monomorphisms, and regular monomorphisms. These translation maps induce well-defined algebraic products on isomorphism classes of graphs, recovering various known graph products and yielding new ones. The first part of the thesis establishes these constructions in the context of simple graphs, then extends them to sets equipped with multiple binary relations simultaneously. This generality forms the structural foundation for the subsequent chapters. While parts of the results appeared in Graph Profiles as Characters on Bicommutative Hopf Algebras (Diehl, Caudillo, Ebrahimi-Fard, Verri, Graphs and Combinatorics, 2025), the thesis provides a significantly broader and more detailed treatment, including extensions not present in the published work. Building on this categorical–algebraic framework, the latter part of the thesis applies these ideas to permutation pattern counting, demonstrating how the developed machinery enlarges the class of permutation patterns for which efficient enumeration techniques are available. Through these contributions, the thesis advances the understanding of combinatorial Hopf algebra structures, deepens the interplay between category theory and combinatorics, and introduces new tools for the analysis of graph and permutation statistics.