In this paper we study enumeration problems for polytopes arising from combinatorial optimization problems. While these polytopes turn out to be quickly intractable for enumeration algorithms designed for general polytopes, tailor-made algorithms using their rich combinatorial features can exhibit strong performances. The main engine of these combinatorial algorithms is the use of the large sym...

A combinatorial map is a connected topological graph cellularly embeddedin a surface. This monograph concentrates on the automorphism groupof a map, which is related to the automorphism groups of a Klein surfaceand a Smarandache manifold, also applied to the enumeration of unrootedmaps on orientable and non-orientable surfaces. A number of results for theautomorphism groups ...

We decribe various aspects of the Al-Salam-Chihara q-Laguerre polynomials. These include combinatorial descriptions of the polynomials, the moments, the orthogonality relation and a combinatorial interpretation of the linearization coefficients. It is remarkable that the corresponding moment sequence appears also in the recent work of Postnikov and Williams on enumeration of totally positive Gr...

In this paper systems of functional equations in infinitely many variables arising in combinatorial enumeration problems are studied. We prove sufficient conditions under which the combinatorial random variables encoded in the generating functions of the system tend to an infinite dimensional Gaussian limiting distribution.

We give bijective proofs that, when combined with one of the combinatorial proofs of the general ballot formula, constitute a combinatorial argument yielding the number of lattice paths from (0, 0) to (n, rn) that touch or cross the diagonal y = rx at exactly k lattice points. This enumeration partitions all lattice paths from (0, 0) to (n, rn). While the resulting formula can be derived using ...

This booklet develops in nearly 200 pages the basics of combinatorial enumeration through an approach that revolves around generating functions. The major objects of interest here are words, trees, graphs, and permutations, which surface recurrently in all areas of discrete mathematics. The text presents the core of the theory with chapters on unlabelled enumeration and ordinary generating func...

The well-known vertex enumeration problem calls for generating all vertices of a polyhedron given by its description as a system of linear inequalities. Recently, a number of combinatorial techniques have been developed and applied successfully to a large number of monotone generation problems from different areas. We consider several such techniques and give examples in which they are applicab...