In-depth combinatorial analysis

The notion of a set partition is exactly that of an equivalence relation, which is well known to everyone. Here, in the case of a finite set N with n elements, the number of partitions of N into k blocks (non-empty parts), or, if we prefer, the number of k-class equivalence relations on N, noted S(n,k), is none other than the famous Stirling number of the second kind. These S(n,k) numbers are used just about everywhere, in algebra, analysis, probability, statistics, etc. We'll be taking a particularly detailed look at them here.

The notion of partitioning an integer n is more theoretical in nature. It is, so to speak, a gigantic generalization of the famous problem of exchanging money: in how many ways can an amount of n francs be realized with coins of 1, 2 and 5 francs? Without integer series, we'd get nowhere, as Euler has shown. In its generality, this theory is at least as much about arithmetic as it is about combinatorics, the latter aspect being the only one considered here.

Finally, the notion of permutation (of a finite set) is taken up again in great detail, and provides an opportunity to introduce such combinatorially fundamental numbers as the Stirling numbers of the first kind s(n,k), the Eulerian numbers A(n,k) which count the permutations of $\left[1,\mathrm{\hspace{0.17em}}n\right]$ by ascending, the tangent numbers a _2n+1 , Taylor coefficients of the integer series expansion of : $\tan \left(x\right)={\displaystyle \sum _{n\ge 0}{a}_{2n+1}\frac{x^{2n+1}}{\left(2n+1\right)\mathrm{ }\mathrm{!}},}$

which count the alternating permutations of $\left[1,\mathrm{\hspace{0.17em}}n\right]$ , etc.

Combinatorial analysis" is the subject of several articles:

• [AF 200] "Elementary combinatorial analysis" ;

• [AF 201] "Advanced...