Algebraic enumeration

Abstract: In this Paper, we illustrate a method (called the ECO method) for enumerating some classes of combinatorial objects. The basic idea of this method is the following: by means of an operator that performs a "local expansion" on the objects, we give some recursive constructions of these classes. We use these constructions to deduce some new funtional equations verified by classes' generating functions. By solving the functional equations, we enumerate the combinatorial objects according to various parameters. We show some applications of the method referring to some classical combinatorial objects, such as: trees, paths, polyminoes and permutations

Abstract: The computational complexity of problems relating to the enumeration of all the vertices of a convex polyhedron defined by linear inequalities is examined. Several published approaches are evaluated in this light. A new algorithm is described, and shown to have a better complexity estimate than existing methods. Empirical evidence supporting the theoretical superiority is presented. Finally vertex enumeration is discussed when the space containing the polyhedra is of fixed dimension and only the size of the inequality system is permitted to vary.

Abstract: This report is part of a series whose aim is to present in a synthetic way the major methods and models in analytic combinatorics. Here, we detail the case of rational and algebraic functions and discuss systematically closure properties, the location of singularities, and consequences regarding combinatorial enumeration. The theory is applied to regular and context-free languages, finite state models, paths in graphs, locally constrained permutati- ons, lattice paths and walks, trees, and planar maps.