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# Algebraic 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. [1]

82 relations: Abstract algebra, Affine space, Alain Lascoux, Alfred Young, Algebraic graph theory, American Mathematical Society, Association scheme, Benz plane, Bernd Sturmfels, Binary relation, Cambridge University Press, Character theory, Coding theory, Combinatorial commutative algebra, Combinatorial design, Combinatorial optimization, Combinatorics, Commutative algebra, Complement graph, Complete graph, Doron Zeilberger, Enumerative combinatorics, Euclidean geometry, Ferdinand Georg Frobenius, Finite field, Finite geometry, Finite set, Galois geometry, General linear group, Geometry, Gian-Carlo Rota, Gilbert de Beauregard Robinson, Graduate Texts in Mathematics, Graph theory, Group (mathematics), Group action, Group representation, Group theory, Integer, Inversive geometry, Isomorphism, Journal of Algebraic Combinatorics, Laguerre plane, Lattice (order), Linear algebra, Linear independence, Marcel-Paul Schützenberger, Mathematician, Mathematics, Mathematics Subject Classification, ... Expand index (32 more) »

## Abstract algebra

In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures.

## Affine space

In mathematics, an affine space is a geometric structure that generalizes the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related to parallelism and ratio of lengths for parallel line segments.

## Alain Lascoux

Alain Lascoux (October 17, 1944 – October 20, 2013) was a French mathematician at the University of Marne la Vallée and Nankai University.

## Alfred Young

Alfred Young, FRS (16 April 1873 – 15 December 1940) was a British mathematician.

## Algebraic graph theory

Algebraic graph theory is a branch of mathematics in which algebraic methods are applied to problems about graphs.

## American Mathematical Society

The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs.

## Association scheme

The theory of association schemes arose in statistics, in the theory of experimental design for the analysis of variance.

## Benz plane

In mathematics, a Benz plane is a type of 2-dimensional geometrical structure, named after the German mathematician Walter Benz.

## Bernd Sturmfels

Bernd Sturmfels (born March 28, 1962 in Kassel, West Germany) is a Professor of Mathematics and Computer Science at the University of California, Berkeley and is a director of the Max Planck Institute for Mathematics in the Sciences in Leipzig since 2017.

## Binary relation

In mathematics, a binary relation on a set A is a set of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2.

## Cambridge University Press

Cambridge University Press (CUP) is the publishing business of the University of Cambridge.

## Character theory

In mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix.

## Coding theory

Coding theory is the study of the properties of codes and their respective fitness for specific applications.

## Combinatorial commutative algebra

Combinatorial commutative algebra is a relatively new, rapidly developing mathematical discipline.

## Combinatorial design

Combinatorial design theory is the part of combinatorial mathematics that deals with the existence, construction and properties of systems of finite sets whose arrangements satisfy generalized concepts of balance and/or symmetry.

## Combinatorial optimization

In applied mathematics and theoretical computer science, combinatorial optimization is a topic that consists of finding an optimal object from a finite set of objects.

## Combinatorics

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.

## Commutative algebra

Commutative algebra is the branch of algebra that studies commutative rings, their ideals, and modules over such rings.

## Complement graph

In graph theory, the complement or inverse of a graph is a graph on the same vertices such that two distinct vertices of are adjacent if and only if they are not adjacent in.

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## Doron Zeilberger

Doron Zeilberger (דורון ציילברגר, born 2 July 1950 in Haifa, Israel) is an Israeli mathematician, known for his work in combinatorics.

## Enumerative combinatorics

Enumerative combinatorics is an area of combinatorics that deals with the number of ways that certain patterns can be formed.

## Euclidean geometry

Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements.

## Ferdinand Georg Frobenius

Ferdinand Georg Frobenius (26 October 1849 – 3 August 1917) was a German mathematician, best known for his contributions to the theory of elliptic functions, differential equations, number theory, and to group theory.

## Finite field

In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements.

## Finite geometry

A finite geometry is any geometric system that has only a finite number of points.

## Finite set

In mathematics, a finite set is a set that has a finite number of elements.

## Galois geometry

Galois geometry (so named after the 19th century French Mathematician Évariste Galois) is the branch of finite geometry that is concerned with algebraic and analytic geometry over a finite field (or Galois field).

## General linear group

In mathematics, the general linear group of degree n is the set of invertible matrices, together with the operation of ordinary matrix multiplication.

## Geometry

Geometry (from the γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.

## Gian-Carlo Rota

Gian-Carlo Rota (April 27, 1932 &ndash; April 18, 1999) was an Italian-born American mathematician and philosopher.

## Gilbert de Beauregard Robinson

Gilbert de Beauregard Robinson (1906–1992) was a Canadian mathematician most famous for his work on combinatorics and representation theory of the symmetric groups, including the Robinson-Schensted algorithm.

## Graph theory

In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.

## Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.

## Group action

In mathematics, an action of a group is a formal way of interpreting the manner in which the elements of the group correspond to transformations of some space in a way that preserves the structure of that space.

## Group representation

In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication.

## Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.

## Integer

An integer (from the Latin ''integer'' meaning "whole")Integer&#x2009;'s first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").

## Inversive geometry

In geometry, inversive geometry is the study of those properties of figures that are preserved by a generalization of a type of transformation of the Euclidean plane, called inversion.

## Isomorphism

In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism.

## Journal of Algebraic Combinatorics

Journal of Algebraic Combinatorics is a peer-reviewed scientific journal covering algebraic combinatorics.

## Laguerre plane

In mathematics, a Laguerre plane is one of the Benz planes: M&ouml;bius plane, Laguerre plane and Minkowski plane, named after the French mathematician Edmond Nicolas Laguerre.

## Lattice (order)

A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra.

## Linear algebra

Linear algebra is the branch of mathematics concerning linear equations such as linear functions such as and their representations through matrices and vector spaces.

## Linear independence

In the theory of vector spaces, a set of vectors is said to be if one of the vectors in the set can be defined as a linear combination of the others; if no vector in the set can be written in this way, then the vectors are said to be.

## Marcel-Paul Schützenberger

Marcel-Paul "Marco" Schützenberger (October 24, 1920 – July 29, 1996) was a French mathematician and Doctor of Medicine.

## Mathematician

A mathematician is someone who uses an extensive knowledge of mathematics in his or her work, typically to solve mathematical problems.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## Mathematics Subject Classification

The Mathematics Subject Classification (MSC) is an alphanumerical classification scheme collaboratively produced by staff of, and based on the coverage of, the two major mathematical reviewing databases, Mathematical Reviews and Zentralblatt MATH.

## Matroid

In combinatorics, a branch of mathematics, a matroid is a structure that abstracts and generalizes the notion of linear independence in vector spaces.

## Möbius plane

In mathematics, a Möbius plane (named after August Ferdinand Möbius) is one of the Benz planes: Möbius plane, Laguerre plane and Minkowski plane.

## Melvin Hochster

Melvin Hochster (born August 2, 1943) is an eminent American mathematician, regarded as one of the leading commutative algebraists active today.

## Neighbourhood (graph theory)

In graph theory, an adjacent vertex of a vertex v in a graph is a vertex that is connected to v by an edge.

## Network theory

Network theory is the study of graphs as a representation of either symmetric relations or asymmetric relations between discrete objects.

## Non-Desarguesian plane

In mathematics, a non-Desarguesian plane, named after Girard Desargues, is a projective plane that does not satisfy Desargues' theorem, or in other words a plane that is not a Desarguesian plane.

## Partially ordered set

In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set.

## Percy Alexander MacMahon

Percy Alexander MacMahon (born 26 September 1854, Sliema, British Malta &ndash; 25 December 1929, Bognor Regis, England) was a mathematician, especially noted in connection with the partitions of numbers and enumerative combinatorics.

## Pixel

In digital imaging, a pixel, pel, dots, or picture element is a physical point in a raster image, or the smallest addressable element in an all points addressable display device; so it is the smallest controllable element of a picture represented on the screen.

## Point (geometry)

In modern mathematics, a point refers usually to an element of some set called a space.

## Polyhedral 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.

## Polytope

In elementary geometry, a polytope is a geometric object with "flat" sides.

## Projective plane

In mathematics, a projective plane is a geometric structure that extends the concept of a plane.

## Projective space

In mathematics, a projective space can be thought of as the set of lines through the origin of a vector space V. The cases when and are the real projective line and the real projective plane, respectively, where R denotes the field of real numbers, R2 denotes ordered pairs of real numbers, and R3 denotes ordered triplets of real numbers.

## Regular graph

In graph theory, a regular graph is a graph where each vertex has the same number of neighbors; i.e. every vertex has the same degree or valency.

## Representation theory

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures.

## Representation theory of the symmetric group

In mathematics, the representation theory of the symmetric group is a particular case of the representation theory of finite groups, for which a concrete and detailed theory can be obtained.

## Richard P. Stanley

Richard Peter Stanley (born June 23, 1944 in New York City, New York) is the Norman Levinson Professor of Applied Mathematics at the Massachusetts Institute of Technology, in Cambridge, Massachusetts.

## Ring of symmetric functions

In algebra and in particular in algebraic combinatorics, the ring of symmetric functions is a specific limit of the rings of symmetric polynomials in n indeterminates, as n goes to infinity.

## Schubert calculus

In mathematics, Schubert calculus is a branch of algebraic geometry introduced in the nineteenth century by Hermann Schubert, in order to solve various counting problems of projective geometry (part of enumerative geometry).

## Strongly regular graph

In graph theory, a strongly regular graph is defined as follows.

## Symmetric function

In mathematics, a symmetric function of n variables is one whose value given n arguments is the same no matter the order of the arguments.

## Symmetric group

In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions.

## Symmetric polynomial

In mathematics, a symmetric polynomial is a polynomial in variables, such that if any of the variables are interchanged, one obtains the same polynomial.

## Symmetry in mathematics

Symmetry occurs not only in geometry, but also in other branches of mathematics.

## The Princeton Companion to Mathematics

The Princeton Companion to Mathematics is a book, edited by Timothy Gowers with associate editors June Barrow-Green and Imre Leader, and published in 2008 by Princeton University Press.

## Topology

In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.

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## University of Cambridge

The University of Cambridge (informally Cambridge University)The corporate title of the university is The Chancellor, Masters, and Scholars of the University of Cambridge.

## Vector space

A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.

## W. V. D. Hodge

Sir William Vallance Douglas Hodge FRS FRSE (17 June 1903 – 7 July 1975) was a British mathematician, specifically a geometer.

## Young tableau

In mathematics, a Young tableau (plural: tableaux) is a combinatorial object useful in representation theory and Schubert calculus.

## References

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