Discrete Mathematics

TL;DR

This paper offers concise teaching notes for a 30-hour introductory course on Enumerative Algebraic Combinatorics, emphasizing elementary derivation techniques and combinatorial principles without requiring advanced computation.

Contribution

It introduces a self-contained, combinatorial approach to teaching Enumerative Algebraic Combinatorics, focusing on elementary methods like the bad element and overcounting principles.

Findings

Provides a sequence of derivations from basic principles.

Uses elementary combinatorial methods for formula derivation.

Avoids complex computations in proofs.

Abstract

The purpose of the present work is to provide short and supple teaching notes for a $30$ hours introductory course on elementary \textit{Enumerative Algebraic Combinatorics}. We fully adopt the \textit{Rota way}. The themes are organized into a suitable sequence that allows us to derive any result from the preceding ones by elementary processes. Definitions of \textit{combinatorial coefficients} are just by their \textit{combinatorial meaning}. The derivation techniques of formulae/results are founded upon constructions and two general and elementary principles/methods: - The \textit{bad element} method (for \textit{recursive} formulae). As the reader should recognize, the bad element method might be regarded as a combinatorial companion of the idea of \textit{conditional probability}. - The \textit{overcounting} principle (for \textit{close form} formulae). Therefore, \textit{no computation} is required in \textit{proofs}: \textit{computation formulae are byproducts of combinatorial constructions}. We tried to provide a self-contained presentation: the only prerequisite is standard high school mathematics. We limited ourselves to the \textit{combinatorial point of view}: we invite the reader to draw the (obvious) \textit{probabilistic interpretations}.