Young tableau

A Young tableau is a filling of the cells of a partition diagram with entries subject to ordering or other rules. Tableaux encode representations of symmetric groups and organize many counting identities.

Integer partition: A way to write a positive integer as a sum of positive integers, with order disregarded. Its parts specify the row lengths of a tableau’s diagram.
Robinson–Schensted correspondence: A bijection between permutations and pairs of standard Young tableaux of the same shape. It converts permutation structure into a pair of tableaux.
Schensted insertion: An algorithm that inserts a word into a tableau by replacing entries and moving displaced entries to later rows. It constructs the insertion tableau in the Robinson–Schensted correspondence.
Symmetric group: The group of all permutations of a finite set, with composition as its operation. Its irreducible representations are indexed by partitions and modeled using tableaux.
Shifted Young tableau: A filling of a shifted diagram associated with a strict partition and subject to tableau ordering rules. Its shifted shape and rules differ from ordinary partition tableaux.
Ferrers diagram: A left-justified array of cells whose row lengths are given by an integer partition. It supplies the cells that a Young tableau fills.
Schur function: A symmetric polynomial indexed by a partition and expressible through tableau-generating sums. Semistandard tableaux provide a direct formula for its coefficients.
Jeu de taquin: A sliding procedure on skew tableaux that moves an empty cell through neighboring entries. Its slides transform skew shapes while preserving key tableau information.
Representation of a finite group: A realization of a finite group as linear transformations of a vector space. Young tableaux provide combinatorial bases and dimension counts for key representations.
Increasing tableau: A filling of a partition diagram by positive integers that strictly increase along rows and columns. It allows entries beyond the standard range while requiring strict increase.

Linked from 3 pages

Algebraic combinatoricsBroader topic: Tableaux encode representations, permutation statistics, and identities for symmetric functions.