Algebraic combinatorics : walks, trees, tableaux, and more / Richard P. Stanley

This book is intended to be a one-semester textbook for undergraduates. So its title seems a little bit too general. As the subtitle suggests, the eleven first chapters are brief introductions to some aspects of the theory of trees and graphs essentially using linear algebra. The twelth one is a potpourri of short problems of various kinds (counting, probability, algebraic number theory).

The first chapter counts the walks on a graph with the associated matrix. The second one specializes to the n-cube and the Radon transform. The third one is about random walks. From the fourth one on, posets are introduced, the Sperner property is studied, then Boolean algebras and the quotient poset by a subgroup of the symmetric group. Young diagrams and Polya’s theory of enumeration (coloring of combinatorial objects) follow in 6 and 7. Some formulas on Young tableaux are derived in Chapter 8, then in 9 the matrix-tree theorem counts the number of spanning trees of a graph. Eulerian tours lead to the notions of Eulerian digraph and oriented trees in Chapter 10. The overview of graph theory ends with some clues about electrical networks in Chapter 11. Each chapter is short enough and forms a “lesson” in some sense. They are often followed by notes and complements, and always end with a collection of exercices (some hints at the end of the book). Many figures illustrate the notions and the properties as they are introduced. (zbMath)