ABSTRACT
I. Introduction 121II. General Sources 121A. General texts 121B. Source books, surveys, and general handbooks 123C. Online tools and pathfinders 123D. Major journals and article sources 124III. Enumerative Combinatorics 125A. General texts 126B. Exact enumerations and generating functions 126C. Asymptotics 127D. q-Series, number partitions, hypergeometric series and identities 127E. Special functions, orthogonal polynomials, and hypergeometric transformations 127F. Combinatorics and statistical physics 128IV. Graph Theory 129A. History and historically important works 129B. General texts 129C. Surveys and applications 130D. Topological graph theory 131E. Algebraic graph theory and matrix theory 131F. Coloring of graphs and hypergraphs 132G. Perfect graphs 133 119
H. Directed graphs 133I. Enumeration of graphs and maps 134J. Hypergraphs 134K. Random graphs 134L. Matching theory and network flows 134V. Designs and Codes 135A. General sources 135B. Coding theory 136C. Design theory 136D. Incidence geometries, buildings,generalized polygons 136E. Latin squares, orthogonal arrays,Hadamard matrices 137VI. Extremal Combinatorics 137A. Extremal set theory 137B. Probabilistic methods 138C. Ramsey theory 138VII. Partially Ordered Sets and Lattices 138A. Historically important 139B. General texts 139C. Surveys 139VIII. Algebraic and Topological Combinatorics 139A. General sources 139B. Representation theory, symmetric functions,and the symmetric group 140C. Combinatorial commutative algebraand algebraic geometry 140D. Topological combinatorics 142IX. Discrete Geometry, Convexity, and Optimization 142A. General sources 142B. Packing, covering, and tiling 143C. Combinatorial and discrete geometry 143D. Polytopes and polyhedra 144E. Valuations and Brunn-Minkowski theory 145F. Linear and integer programming 145G. Matroids and oriented matroids 146H. Computational geometry 147
Combinatorics is the study of discrete structures in mathematics, often dealing with finite sets, but sometimes the infinite as well. Combinatorics as a subject has the allure of many easy-to-state but challenging problems. Combinatorics problems often arise in almost any area of mathematics, sometimes as the end product of the reduction of some seemingly continuous or infinite problem to something finite. Combinatorics came of age during the nineteenth and twentieth centuries, passing from a seemingly disparate bag of tricks for solving problems that sound like brain-teasers into a body of theoretical tools and frameworks within which to classify and understand problems.This chapter recommends resources for learning about the main areas and topics of combinatorics. Inevitably we haven’t done justice to certain areas: some notable ones that we intentionally omitted are combinatorial number theory, combinatorial game theory and applications to economics, combinatorial aspects of the theory of algorithms and theoretical computer science, combinatorics of languages, and cellular automata. II. GENERAL SOURCES
NL Biggs, EK Lloyd, and RJ Wilson. History of combinatorics. In: Graham, Grotschefi and Lovasz, eds. Handbook of Combinatorics. Cambridge, MA: MIT Press, 1995, pp 2163-2198.E Netto. Lehrbuch der Kombinatorik. Leipzig: Teubner, 1901.PA MacMahon. Combinatory Analysis. Cambridge: Cambridge University Press, 1915-1916. General Combinatorics Texts
M Aigner. Combinatorial Theory. Grundlehren der mathematischen Wissenschaften, Vol. 234. Berlin: Springer, 1979.Graduate level. VK Balakrishnan. Schaum’s Outline of Theory and Problems of Combinatorics. New York: McGraw-Hill, 1995.Undergraduate level. EA Bender and SG Williamson. Foundations of Applied Combinatorics. Redwood City, CA: Addison-Wesley, 1991.Undergraduate level.
