Elementary combinatorial analysis

Combinatorial analysis is a branch of mathematics that deals with problems of enumeration (or counting), enumeration (or listing) and estimation (framing and asymptotism) on finite sets.

This vision, while admittedly rather reductive, is nonetheless very rich. In the abundance of so-called combinatorial subjects, we've had to make a choice in this article, and exclude certain related and important theories, such as graph theory, for example. The main applications of the subject are obviously in probability calculus and statistics. Nevertheless, we must not conceal the fact that many of the traditional problems of analysis, algebra and geometry are combinatorial in nature, and of course, even more so, those recently posed by computer science.

This science of Combinatorial Analysis is said to have originated in France with the work of Pascal, who, faced with questions of probability in games, was probably one of the first to give the coefficients of the development of the binomial (x + y) ⁿ by means of his triangle, which he then called the "mystic triangle". But many other scientists of the XVII ^e century contributed to the nascent edifice. Among them were Leibniz, Newton, Wallis, Jacques Bernoulli and Moivre... After that, the XVIII ^e and XIX ^e centuries were sparse in works on the subject, and the science seemed somewhat neglected. At the beginning of the XX ^e century, the work of Netto (Germany), MacMahon (England) and André and Lucas (France) gradually revitalized this discipline, which finally came into its own in the 1950s.

The very title of this specialty has itself fluctuated over time. From the classic "Combinatorial Analysis" we have moved on to "Combinatorics", a pleasant and convenient condensation. But we also say "Combinatorics", from the German Kombinatorik, the title of Netto's famous 1901 book, also used in English as Combinatorics...

And what should we call those whose job it is to research (and sometimes even find!) Combinatorics? Surely, the French language would have them be called "Combinatoriens"... Don't we have Histoire → Historien, Oratoire → Oratorien, Prétoire → Prétorien? But some authorized people still prefer "Combinatorialistes", like Mémoire → Mémorialiste, or even, more rarely, "Combinatoriciens", like Informatique → Informaticien... To each his own!

The methods of the Combinatorians, which were originally adapted solely to the solution of particular problems, are now tending to use general methods of solution: generating functions,...