Alguns problemas mais elaborados em análise combinatória elementar, I: Enumeração de funções

In the study of elementary combinatorics, we are introduced to the principles of addition and multiplication and their most immediate applications: solving counting problems involving permutations and combinations with or without repetitions. However, variations of these problems are frequently raised by students in class that are often not covered in the most widely used textbooks or undergraduate syllabi. In this work, which corresponds to the first part of a two-part exposition, we provide a brief but thorough review of the basic counting principles, including the inclusion-exclusion principle, and explore the counting of functions and of increasing functions. To count surjective functions, we introduce Stirling numbers of the second kind and explore several interesting identities and problems related to these numbers. Finally, we mention the classification of the fundamental combinatorial problems in terms of the possible distributions of balls in urns, distinct or identical, subject to different constraints, which became known as the "twelvefold way." The work contains several observations that offer historical notes and address related topics that we believe will be of interest to the readers.