Analyse combinatoire

Combinatorial analysis is the art of enumeration, a branch of discrete mathematics that counts combinatorial structures derived from finite sets.

The first chapters introduce the essential concepts: usual configurations (combinations, arrangements, etc.), generating series (ordinary or exponential), inclusion-exclusion principle (sieve formula). These fundamental tools enable us to establish classic results (number of overjections, disturbances, etc.) and lead to the study of remarkable number sequences, such as Fibonacci or Bernoulli.

Subsequent chapters deal with more elaborate topics at the heart of combinatorics: integer partitions, set partitions (Bell numbers, Stirling numbers), permutations (alternating, with fixed points, Pólya theory...), graph theory (couplings, spanning trees...), partially ordered sets, and more. A variety of topics are covered: specific partitions (spaced, uncrossed, singleton-free, etc.), parentheses, trees (ordered, binary, bushes, etc.), Dyck words, Delannoy paths, etc., bringing out new sequences of integers: Catalan numbers, Motzkin numbers, Riordan numbers, Narayana numbers, etc. Each chapter contains corrected exercises, applications or extensions of the course.

This book is aimed at university and engineering school students, doctoral candidates, teachers, researchers, engineers and, more generally, anyone wishing to delve deeper into this subject. It assumes a certain ease with general mathematics at undergraduate level, but does not require any prior knowledge of combinatorics.