Overview of some general results in combinatorial enumeration

TL;DR

This survey reviews broad results in combinatorial enumeration, focusing on growth properties of hereditary structures and five key topics including lattice point counting, language growth, and regular graph enumeration.

Contribution

It compiles and discusses general results and recent advances across multiple areas of combinatorial enumeration, highlighting five specific topics.

Findings

Growth patterns of hereditary combinatorial properties

Enumeration techniques for lattice points and languages

Asymptotic behaviors and periodicity in combinatorial structures

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 five 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, 4. frequent occurrence of the asymptotics cn^{-3/2}r^n and 5. ultimate modular periodicity of numbers of MSOL-definable structures.