A unifying account of enumerative combinatorics in which counting problems are solved by attaching an algebraic or geometric structure to the objects being counted and then reading the answer off that structure. The four toolkits are: (1) generating functions, where a sequence becomes an element of the formal power series ring and the sum, product and composition rules turn combinatorial decompositions into algebra, with rational, algebraic and D-finite series corresponding to c-recursive and P-recursive sequences; (2) linear algebra, where determinants and Pfaffians count spanning trees (Matrix-Tree), Eulerian circuits (BEST), non-intersecting lattice paths (Lindstrom-Gessel-Viennot) and dimer coverings (Kasteleyn); (3) posets, where the Mobius function of an incidence algebra inverts a counting relation and the order complex ties enumeration to topology, producing zeta and order polynomials, flag f- and h-vectors and the cd-index; and (4) discrete geometry, where convex polytopes, their triangulations, f- and h-vectors, Ehrhart reciprocity, hyperplane arrangements and their characteristic polynomials, and matroids with the universal Tutte polynomial each convert an enumeration problem into a structural one. The recurring lesson is that a well-chosen structure both computes a number and explains it, so that reciprocity theorems, unimodality results and bijections appear as consequences rather than coincidences.
Ardila's 144-page Chapter 1 of the Handbook of Enumerative Combinatorics: a survey of two toolkits for counting combinatorial objects. Part 1 (algebraic methods): generating functions -- formal power…