Abstract

This survey article is devoted to general results in combinatorial enumeration. The first part surveys results on growth of hereditary properties of combinatorial structures. These include permutations, ordered and unordered graphs and hypergraphs, relational structures, and others. The second part advertises four topics in general enumeration: 1. counting lattice points in lattice polytopes, 2. growth of context-free languages, 3. holonomicity (i.e., P-recursiveness) of numbers of labeled regular graphs and 4. ultimate modular periodicity of numbers of MSOL-definable structures.

Introduction

We survey some general results in combinatorial enumeration. A problem in enumeration is (associated with) an infinite sequence P = (S1, S2, …) of finite sets Si. Its counting function fP is given by fP (n) = |Sn|, the cardinality of the set Sn. We are interested in results of the following kind on general classes of problems and their counting functions.

Scheme of general results in combinatorial enumeration. The counting function fP of every problem P in the class C belongs to the class of functions F. Formally, {fP | P ∈ C} ⊂ F.

The larger C is, and the more specific the functions in F are, the stronger the result. The present overview is a collection of many examples of this scheme.