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  "body": "\n## Definition\n\n*Algebraic combinatorics* is the branch of mathematics with two complementary aims:\n\n1. to solve problems of [[combinatorics|0JKW7W7XFR]] with tools of abstract algebra, such as groups, [[group actions|0WCY42EV9V]], rings, linear algebra and representation theory;\n2. to describe algebraic objects, such as bases, structure constants, characters and dimensions, by combinatorial rules that count explicit finite structures.\n\n## Remarks\n\nExample of aim 1: the number of [[orbits|0YQPZB5Q1G]] of a group acting on colourings is computed by [[Burnside's lemma|0Z47M1B2VS]] and the [[Pólya enumeration theorem|MPYQ39P01F]].\nExample of aim 2: the product of two [[Schur polynomials|43VFMJCDYJ]] is a sum of Schur polynomials.\nThe [[Littlewood–Richardson rule|Q490XBH1Q2]] gives the coefficients as numbers of certain tableaux.\n\nNon-example: a proof that a graph with a property exists, by showing that a random graph has it with positive probability, meets neither aim.\nIt belongs to the [[probabilistic method|3B55HNS754]].\n\nThe main objects include [[symmetric functions|PAJEHWXW6M]], [[Young tableaux|Q6TH9PH2R7]], [[association schemes|C338GKB4VJ]] and [[incidence algebras|1H9SFWHYEB]] of posets.\n",
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