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Some general results in combinatorial enumeration

Published online by Cambridge University Press:  05 October 2010

By
Martin Klazar
Edited by
Steve Linton ,
Nik Ruškuc  and
Vincent Vatter
Martin Klazar
Affiliation:
Department of Applied Mathematics Charles University 118 00 Praha Czech Republic
Steve Linton
Affiliation:
University of St Andrews, Scotland
Nik Ruškuc
Affiliation:
University of St Andrews, Scotland
Vincent Vatter
Affiliation:
Dartmouth College, New Hampshire
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Summary

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.

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Permutation Patterns , pp. 3 - 40
Publisher: Cambridge University Press
Print publication year: 2010

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