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**Combinatorics** is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science.

- Definition
- History
- Approaches and subfields of combinatorics
- Enumerative combinatorics
- Analytic combinatorics
- Partition theory
- Graph theory
- Design theory
- Finite geometry
- Order theory
- Matroid theory
- Extremal combinatorics
- Probabilistic combinatorics
- Algebraic combinatorics
- Combinatorics on words
- Geometric combinatorics
- Topological combinatorics
- Arithmetic combinatorics
- Infinitary combinatorics
- Related fields
- Combinatorial optimization
- Coding theory
- Discrete and computational geometry
- Combinatorics and dynamical systems
- Combinatorics and physics
- See also
- Notes
- References
- External links

Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry,^{ [1] } as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an *ad hoc* solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right.^{ [2] } One of the oldest and most accessible parts of combinatorics is graph theory, which by itself has numerous natural connections to other areas. Combinatorics is used frequently in computer science to obtain formulas and estimates in the analysis of algorithms.

A mathematician who studies combinatorics is called a *combinatorialist*.

The full scope of combinatorics is not universally agreed upon.^{ [3] } According to H.J. Ryser, a definition of the subject is difficult because it crosses so many mathematical subdivisions.^{ [4] } Insofar as an area can be described by the types of problems it addresses, combinatorics is involved with:

- the
*enumeration*(counting) of specified structures, sometimes referred to as arrangements or configurations in a very general sense, associated with finite systems, - the
*existence*of such structures that satisfy certain given criteria, - the
*construction*of these structures, perhaps in many ways, and *optimization*: finding the "best" structure or solution among several possibilities, be it the "largest", "smallest" or satisfying some other*optimality criterion*.

Leon Mirsky has said: "combinatorics is a range of linked studies which have something in common and yet diverge widely in their objectives, their methods, and the degree of coherence they have attained."^{ [5] } One way to define combinatorics is, perhaps, to describe its subdivisions with their problems and techniques. This is the approach that is used below. However, there are also purely historical reasons for including or not including some topics under the combinatorics umbrella.^{ [6] } Although primarily concerned with finite systems, some combinatorial questions and techniques can be extended to an infinite (specifically, countable) but discrete setting.

Basic combinatorial concepts and enumerative results appeared throughout the ancient world. Indian physician Sushruta asserts in Sushruta Samhita that 63 combinations can be made out of 6 different tastes, taken one at a time, two at a time, etc., thus computing all 2^{6} − 1 possibilities. Greek historian Plutarch discusses an argument between Chrysippus (3rd century BCE) and Hipparchus (2nd century BCE) of a rather delicate enumerative problem, which was later shown to be related to Schröder–Hipparchus numbers.^{ [7] }^{ [8] }^{ [9] } Earlier, in the * Ostomachion *, Archimedes (3rd century BCE) may have considered the number of configurations of a tiling puzzle,^{ [10] } while combinatorial interests possibly were present in lost works by Apollonius.^{ [11] }^{ [12] }

In the Middle Ages, combinatorics continued to be studied, largely outside of the European civilization. The Indian mathematician Mahāvīra (c. 850) provided formulae for the number of permutations and combinations,^{ [13] }^{ [14] } and these formulas may have been familiar to Indian mathematicians as early as the 6th century CE.^{ [15] } The philosopher and astronomer Rabbi Abraham ibn Ezra (c. 1140) established the symmetry of binomial coefficients, while a closed formula was obtained later by the talmudist and mathematician Levi ben Gerson (better known as Gersonides), in 1321.^{ [16] } The arithmetical triangle—a graphical diagram showing relationships among the binomial coefficients—was presented by mathematicians in treatises dating as far back as the 10th century, and would eventually become known as Pascal's triangle. Later, in Medieval England, campanology provided examples of what is now known as Hamiltonian cycles in certain Cayley graphs on permutations.^{ [17] }^{ [18] }

During the Renaissance, together with the rest of mathematics and the sciences, combinatorics enjoyed a rebirth. Works of Pascal, Newton, Jacob Bernoulli and Euler became foundational in the emerging field. In modern times, the works of J.J. Sylvester (late 19th century) and Percy MacMahon (early 20th century) helped lay the foundation for enumerative and algebraic combinatorics. Graph theory also enjoyed an increase of interest at the same time, especially in connection with the four color problem.

In the second half of the 20th century, combinatorics enjoyed a rapid growth, which led to establishment of dozens of new journals and conferences in the subject.^{ [19] } In part, the growth was spurred by new connections and applications to other fields, ranging from algebra to probability, from functional analysis to number theory, etc. These connections shed the boundaries between combinatorics and parts of mathematics and theoretical computer science, but at the same time led to a partial fragmentation of the field.

Enumerative combinatorics is the most classical area of combinatorics and concentrates on counting the number of certain combinatorial objects. Although counting the number of elements in a set is a rather broad mathematical problem, many of the problems that arise in applications have a relatively simple combinatorial description. Fibonacci numbers is the basic example of a problem in enumerative combinatorics. The twelvefold way provides a unified framework for counting permutations, combinations and partitions.

Analytic combinatorics concerns the enumeration of combinatorial structures using tools from complex analysis and probability theory. In contrast with enumerative combinatorics, which uses explicit combinatorial formulae and generating functions to describe the results, analytic combinatorics aims at obtaining asymptotic formulae.

Partition theory studies various enumeration and asymptotic problems related to integer partitions, and is closely related to q-series, special functions and orthogonal polynomials. Originally a part of number theory and analysis, it is now considered a part of combinatorics or an independent field. It incorporates the bijective approach and various tools in analysis and analytic number theory and has connections with statistical mechanics. Partitions can be graphically visualized with Young diagrams or Ferrers diagrams. They occur in a number of branches of mathematics and physics, including the study of symmetric polynomials and of the symmetric group and in group representation theory in general.

Graphs are fundamental objects in combinatorics. Considerations of graph theory range from enumeration (e.g., the number of graphs on *n* vertices with *k* edges) to existing structures (e.g., Hamiltonian cycles) to algebraic representations (e.g., given a graph *G* and two numbers *x* and *y*, does the Tutte polynomial *T*_{G}(*x*,*y*) have a combinatorial interpretation?). Although there are very strong connections between graph theory and combinatorics, they are sometimes thought of as separate subjects.^{ [20] } While combinatorial methods apply to many graph theory problems, the two disciplines are generally used to seek solutions to different types of problems.

Design theory is a study of combinatorial designs, which are collections of subsets with certain intersection properties. Block designs are combinatorial designs of a special type. This area is one of the oldest parts of combinatorics, such as in Kirkman's schoolgirl problem proposed in 1850. The solution of the problem is a special case of a Steiner system, which systems play an important role in the classification of finite simple groups. The area has further connections to coding theory and geometric combinatorics.

Combinatorial design theory can be applied to the area of design of experiments. Some of the basic theory of combinatorial designs originated in the statistician Ronald Fisher's work on the design of biological experiments. Modern applications are also found in a wide gamut of areas including finite geometry, tournament scheduling, lotteries, mathematical chemistry, mathematical biology, algorithm design and analysis, networking, group testing and cryptography.^{ [21] }

Finite geometry is the study of geometric systems having only a finite number of points. Structures analogous to those found in continuous geometries (Euclidean plane, real projective space, etc.) but defined combinatorially are the main items studied. This area provides a rich source of examples for design theory. It should not be confused with discrete geometry (combinatorial geometry).

Order theory is the study of partially ordered sets, both finite and infinite. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". Various examples of partial orders appear in algebra, geometry, number theory and throughout combinatorics and graph theory. Notable classes and examples of partial orders include lattices and Boolean algebras.

Matroid theory abstracts part of geometry. It studies the properties of sets (usually, finite sets) of vectors in a vector space that do not depend on the particular coefficients in a linear dependence relation. Not only the structure but also enumerative properties belong to matroid theory. Matroid theory was introduced by Hassler Whitney and studied as a part of order theory. It is now an independent field of study with a number of connections with other parts of combinatorics.

Extremal combinatorics studies how large or how small a collection of finite objects (numbers, graphs, vectors, sets, etc.) can be, if it has to satisfy certain restrictions. Much of extremal combinatorics concerns classes of set systems; this is called extremal set theory. For instance, in an *n*-element set, what is the largest number of *k*-element subsets that can pairwise intersect one another? What is the largest number of subsets of which none contains any other? The latter question is answered by Sperner's theorem, which gave rise to much of extremal set theory.

The types of questions addressed in this case are about the largest possible graph which satisfies certain properties. For example, the largest triangle-free graph on *2n* vertices is a complete bipartite graph *K _{n,n}*. Often it is too hard even to find the extremal answer

Ramsey theory is another part of extremal combinatorics. It states that any sufficiently large configuration will contain some sort of order. It is an advanced generalization of the pigeonhole principle.

In probabilistic combinatorics, the questions are of the following type: what is the probability of a certain property for a random discrete object, such as a random graph? For instance, what is the average number of triangles in a random graph? Probabilistic methods are also used to determine the existence of combinatorial objects with certain prescribed properties (for which explicit examples might be difficult to find) by observing that the probability of randomly selecting an object with those properties is greater than 0. This approach (often referred to as *the* probabilistic method) proved highly effective in applications to extremal combinatorics and graph theory. A closely related area is the study of finite Markov chains, especially on combinatorial objects. Here again probabilistic tools are used to estimate the mixing time.^{[ clarification needed ]}

Often associated with Paul Erdős, who did the pioneering work on the subject, probabilistic combinatorics was traditionally viewed as a set of tools to study problems in other parts of combinatorics. The area recently grew to become an independent field of combinatorics.

Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algebra. Algebraic combinatorics has come to be seen more expansively as an area of mathematics where the interaction of combinatorial and algebraic methods is particularly strong and significant. Thus the combinatorial topics may be enumerative in nature or involve matroids, polytopes, partially ordered sets, or finite geometries. On the algebraic side, besides group and representation theory, lattice theory and commutative algebra are common.

Combinatorics on words deals with formal languages. It arose independently within several branches of mathematics, including number theory, group theory and probability. It has applications to enumerative combinatorics, fractal analysis, theoretical computer science, automata theory, and linguistics. While many applications are new, the classical Chomsky–Schützenberger hierarchy of classes of formal grammars is perhaps the best-known result in the field.

Geometric combinatorics is related to convex and discrete geometry. It asks, for example, how many faces of each dimension a convex polytope can have. Metric properties of polytopes play an important role as well, e.g. the Cauchy theorem on the rigidity of convex polytopes. Special polytopes are also considered, such as permutohedra, associahedra and Birkhoff polytopes. Combinatorial geometry is a historical name for discrete geometry.

It includes a number of subareas such as polyhedral combinatorics (the study of faces of convex polyhedra), convex geometry (the study of convex sets, in particular combinatorics of their intersections), and discrete geometry, which in turn has many applications to computational geometry. The study of regular polytopes, Archimedean solids, and kissing numbers is also a part of geometric combinatorics. Special polytopes are also considered, such as the permutohedron, associahedron and Birkhoff polytope.

Combinatorial analogs of concepts and methods in topology are used to study graph coloring, fair division, partitions, partially ordered sets, decision trees, necklace problems and discrete Morse theory. It should not be confused with combinatorial topology which is an older name for algebraic topology.

Arithmetic combinatorics arose out of the interplay between number theory, combinatorics, ergodic theory, and harmonic analysis. It is about combinatorial estimates associated with arithmetic operations (addition, subtraction, multiplication, and division). Additive number theory (sometimes also called additive combinatorics) refers to the special case when only the operations of addition and subtraction are involved. One important technique in arithmetic combinatorics is the ergodic theory of dynamical systems.

Infinitary combinatorics, or combinatorial set theory, is an extension of ideas in combinatorics to infinite sets. It is a part of set theory, an area of mathematical logic, but uses tools and ideas from both set theory and extremal combinatorics. Some of the things studied include continuous graphs and trees, extensions of Ramsey's theorem, and Martin's axiom. Recent developments concern combinatorics of the continuum ^{ [22] } and combinatorics on successors of singular cardinals.^{ [23] }

Gian-Carlo Rota used the name *continuous combinatorics*^{ [24] } to describe geometric probability, since there are many analogies between *counting* and *measure*.

Combinatorial optimization is the study of optimization on discrete and combinatorial objects. It started as a part of combinatorics and graph theory, but is now viewed as a branch of applied mathematics and computer science, related to operations research, algorithm theory and computational complexity theory.

Coding theory started as a part of design theory with early combinatorial constructions of error-correcting codes. The main idea of the subject is to design efficient and reliable methods of data transmission. It is now a large field of study, part of information theory.

Discrete geometry (also called combinatorial geometry) also began as a part of combinatorics, with early results on convex polytopes and kissing numbers. With the emergence of applications of discrete geometry to computational geometry, these two fields partially merged and became a separate field of study. There remain many connections with geometric and topological combinatorics, which themselves can be viewed as outgrowths of the early discrete geometry.

Combinatorial aspects of dynamical systems is another emerging field. Here dynamical systems can be defined on combinatorial objects. See for example graph dynamical system.

There are increasing interactions between combinatorics and physics, particularly statistical physics. Examples include an exact solution of the Ising model, and a connection between the Potts model on one hand, and the chromatic and Tutte polynomials on the other hand.

- ↑ Björner and Stanley, p. 2
- ↑ Lovász, László (1979).
*Combinatorial Problems and Exercises*. North-Holland. ISBN 978-0821842621. Archived from the original on 2021-04-16. Retrieved 2021-03-23.In my opinion, combinatorics is now growing out of this early stage.

- ↑ Pak, Igor. "What is Combinatorics?". Archived from the original on 17 October 2017. Retrieved 1 November 2017.
- ↑ Ryser 1963 , p. 2
- ↑ Mirsky, Leon (1979), "Book Review" (PDF),
*Bulletin of the American Mathematical Society*, New Series,**1**: 380–388, doi: 10.1090/S0273-0979-1979-14606-8 , archived (PDF) from the original on 2021-02-26, retrieved 2021-02-04 - ↑ Rota, Gian Carlo (1969).
*Discrete Thoughts*. Birkhaüser. p. 50. doi:10.1007/978-0-8176-4775-9. ISBN 978-0-8176-4775-9.... combinatorial theory has been the mother of several of the more active branches of today's mathematics, which have become independent ... . The typical ... case of this is algebraic topology (formerly known as combinatorial topology)

- ↑ Acerbi, F. (2003). "On the shoulders of Hipparchus".
*Archive for History of Exact Sciences*.**57**(6): 465–502. doi:10.1007/s00407-003-0067-0. S2CID 122758966. Archived from the original on 2022-01-23. Retrieved 2021-03-12. - ↑ Stanley, Richard P.; "Hipparchus, Plutarch, Schröder, and Hough",
*American Mathematical Monthly***104**(1997), no. 4, 344–350. - ↑ Habsieger, Laurent; Kazarian, Maxim; Lando, Sergei (1998). "On the Second Number of Plutarch".
*The American Mathematical Monthly*.**105**(5): 446. doi:10.1080/00029890.1998.12004906. - ↑ Netz, R.; Acerbi, F.; Wilson, N. "Towards a reconstruction of Archimedes' Stomachion".
*Sciamvs*.**5**: 67–99. Archived from the original on 2021-04-16. Retrieved 2021-03-12. - ↑ Hogendijk, Jan P. (1986). "Arabic Traces of Lost Works of Apollonius".
*Archive for History of Exact Sciences*.**35**(3): 187–253. doi:10.1007/BF00357307. ISSN 0003-9519. JSTOR 41133783. S2CID 121613986. Archived from the original on 2021-04-18. Retrieved 2021-03-26. - ↑ Huxley, G. (1967). "Okytokion".
*Greek, Roman, and Byzantine Studies*.**8**(3): 203. Archived from the original on 2021-04-16. Retrieved 2021-03-26. - ↑ O'Connor, John J.; Robertson, Edmund F., "Combinatorics",
*MacTutor History of Mathematics Archive*, University of St Andrews - ↑ Puttaswamy, Tumkur K. (2000). "The Mathematical Accomplishments of Ancient Indian Mathematicians". In Selin, Helaine (ed.).
*Mathematics Across Cultures: The History of Non-Western Mathematics*. Netherlands: Kluwer Academic Publishers. p. 417. ISBN 978-1-4020-0260-1. Archived from the original on 2021-04-16. Retrieved 2015-11-15. - ↑ Biggs, Norman L. (1979). "The Roots of Combinatorics".
*Historia Mathematica*.**6**(2): 109–136. doi: 10.1016/0315-0860(79)90074-0 . - ↑ Maistrov, L.E. (1974),
*Probability Theory: A Historical Sketch*, Academic Press, p. 35, ISBN 978-1-4832-1863-2, archived from the original on 2021-04-16, retrieved 2015-01-25. (Translation from 1967 Russian ed.) - ↑ White, Arthur T. (1987). "Ringing the Cosets".
*The American Mathematical Monthly*.**94**(8): 721–746. doi:10.1080/00029890.1987.12000711. - ↑ White, Arthur T. (1996). "Fabian Stedman: The First Group Theorist?".
*The American Mathematical Monthly*.**103**(9): 771–778. doi:10.1080/00029890.1996.12004816. - ↑ See Journals in Combinatorics and Graph Theory Archived 2021-02-17 at the Wayback Machine
- ↑ Sanders, Daniel P.;
*2-Digit MSC Comparison*Archived 2008-12-31 at the Wayback Machine - ↑ Stinson 2003 , pg.1
- ↑ Andreas Blass,
*Combinatorial Cardinal Characteristics of the Continuum*, Chapter 6 in Handbook of Set Theory, edited by Matthew Foreman and Akihiro Kanamori, Springer, 2010 - ↑ Eisworth, Todd (2010), Foreman, Matthew; Kanamori, Akihiro (eds.), "Successors of Singular Cardinals",
*Handbook of Set Theory*, Dordrecht: Springer Netherlands, pp. 1229–1350, doi:10.1007/978-1-4020-5764-9_16, ISBN 978-1-4020-4843-2 , retrieved 2022-08-27 - ↑ "
*Continuous and profinite combinatorics*" (PDF). Archived (PDF) from the original on 2009-02-26. Retrieved 2009-01-03.

**Discrete mathematics** is the study of mathematical structures that can be considered "discrete" rather than "continuous". Objects studied in discrete mathematics include integers, graphs, and statements in logic. By contrast, discrete mathematics excludes topics in "continuous mathematics" such as real numbers, calculus or Euclidean geometry. Discrete objects can often be enumerated by integers; more formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets. However, there is no exact definition of the term "discrete mathematics".

**Combinatorics** is a branch of mathematics concerning the study of finite or countable discrete structures.

**Discrete mathematics** is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not vary smoothly in this way, but have distinct, separated values. Discrete mathematics, therefore, excludes topics in "continuous mathematics" such as calculus and analysis.

**Discrete geometry** and **combinatorial geometry** are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object.

**Lists of mathematics topics** cover a variety of topics related to mathematics. Some of these lists link to hundreds of articles; some link only to a few. The template to the right includes links to alphabetical lists of all mathematical articles. This article brings together the same content organized in a manner better suited for browsing. Lists cover aspects of basic and advanced mathematics, methodology, mathematical statements, integrals, general concepts, mathematical objects, and reference tables. They also cover equations named after people, societies, mathematicians, journals, and meta-lists.

The **Fulkerson Prize** for outstanding papers in the area of discrete mathematics is sponsored jointly by the Mathematical Optimization Society (MOS) and the American Mathematical Society (AMS). Up to three awards of $1,500 each are presented at each (triennial) International Symposium of the MOS. Originally, the prizes were paid out of a memorial fund administered by the AMS that was established by friends of the late Delbert Ray Fulkerson to encourage mathematical excellence in the fields of research exemplified by his work. The prizes are now funded by an endowment administered by MPS.

A **convex polytope** is a special case of a polytope, having the additional property that it is also a convex set contained in the -dimensional Euclidean space . Most texts use the term "polytope" for a bounded convex polytope, and the word "polyhedron" for the more general, possibly unbounded object. Others allow polytopes to be unbounded. The terms "bounded/unbounded convex polytope" will be used below whenever the boundedness is critical to the discussed issue. Yet other texts identify a convex polytope with its boundary.

In the mathematical discipline of graph theory, the **dual graph** of a planar graph G is a graph that has a vertex for each face of G. The dual graph has an edge for each pair of faces in G that are separated from each other by an edge, and a self-loop when the same face appears on both sides of an edge. Thus, each edge e of G has a corresponding dual edge, whose endpoints are the dual vertices corresponding to the faces on either side of e. The definition of the dual depends on the choice of embedding of the graph G, so it is a property of plane graphs rather than planar graphs. For planar graphs generally, there may be multiple dual graphs, depending on the choice of planar embedding of the graph.

**Graduate Texts in Mathematics** (**GTM**) is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standard size. The GTM series is easily identified by a white band at the top of the book.

**Algebraic combinatorics** is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algebra.

The mathematical discipline of **topological combinatorics** is the application of topological and algebro-topological methods to solving problems in combinatorics.

**Polyhedral combinatorics** is a branch of mathematics, within combinatorics and discrete geometry, that studies the problems of counting and describing the faces of convex polyhedra and higher-dimensional convex polytopes.

In discrete geometry and mechanics, **structural rigidity** is a combinatorial theory for predicting the flexibility of ensembles formed by rigid bodies connected by flexible linkages or hinges.

**Polymake** is software for the algorithmic treatment of convex polyhedra.

In geometry and polyhedral combinatorics, an **integral polytope** is a convex polytope whose vertices all have integer Cartesian coordinates. That is, it is a polytope that equals the convex hull of its integer points. Integral polytopes are also called **lattice polytopes** or **Z-polytopes**. The special cases of two- and three-dimensional integral polytopes may be called polygons or polyhedra instead of polytopes, respectively.

Mathematics is a broad subject that is commonly divided in many **areas** that may be defined by their objects of study, by the used methods, or by both. For example, analytic number theory is a subarea of number theory devoted to the use of methods of analysis for the study of natural numbers.

* Convex Polytopes* is a graduate-level mathematics textbook about convex polytopes, higher-dimensional generalizations of three-dimensional convex polyhedra. It was written by Branko Grünbaum, with contributions from Victor Klee, Micha Perles, and G. C. Shephard, and published in 1967 by John Wiley & Sons. It went out of print in 1970. A second edition, prepared with the assistance of Volker Kaibel, Victor Klee, and Günter M. Ziegler, was published by Springer-Verlag in 2003, as volume 221 of their book series Graduate Texts in Mathematics.

**Reverse-search algorithms** are a class of algorithms for generating all objects of a given size, from certain classes of combinatorial objects. In many cases, these methods allow the objects to be generated in polynomial time per object, using only enough memory to store a constant number of objects. They work by organizing the objects to be generated into a spanning tree of their state space, and then performing a depth-first search of this tree.

- Björner, Anders; and Stanley, Richard P.; (2010);
*A Combinatorial Miscellany* - Bóna, Miklós; (2011);
*A Walk Through Combinatorics (3rd ed.)*. ISBN 978-981-4335-23-2 , 978-981-4460-00-2 - Graham, Ronald L.; Groetschel, Martin; and Lovász, László; eds. (1996);
*Handbook of Combinatorics*, Volumes 1 and 2. Amsterdam, NL, and Cambridge, MA: Elsevier (North-Holland) and MIT Press. ISBN 0-262-07169-X - Lindner, Charles C.; and Rodger, Christopher A.; eds. (1997);
*Design Theory*, CRC-Press. ISBN 0-8493-3986-3. - Riordan, John (2002) [1958],
*An Introduction to Combinatorial Analysis*, Dover, ISBN 978-0-486-42536-8 - Ryser, Herbert John (1963),
*Combinatorial Mathematics*, The Carus Mathematical Monographs(#14), The Mathematical Association of America - Stanley, Richard P. (1997, 1999);
*Enumerative Combinatorics*, Volumes 1 and 2, Cambridge University Press. ISBN 0-521-55309-1 , 0-521-56069-1 - Stinson, Douglas R. (2003),
*Combinatorial Designs: Constructions and Analysis*, New York: Springer, ISBN 0-387-95487-2 - van Lint, Jacobus H.; and Wilson, Richard M.; (2001);
*A Course in Combinatorics*, 2nd ed., Cambridge University Press. ISBN 0-521-80340-3

- "Combinatorial analysis",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994] - Combinatorial Analysis – an article in Encyclopædia Britannica Eleventh Edition
- Combinatorics, a MathWorld article with many references.
- Combinatorics, from a
*MathPages.com*portal. - The Hyperbook of Combinatorics, a collection of math articles links.
- The Two Cultures of Mathematics by W.T. Gowers, article on problem solving vs theory building.
- "Glossary of Terms in Combinatorics"
- List of Combinatorics Software and Databases

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