= Young tableau
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A Young tableau is a filling of a <Young diagram> with entries, typically positive <integers>. The shape is the <partition of an integer> giving its row lengths. We draw <Young tableaux> in the English convention of <Young diagram>, with left-aligned rows arranged from top to bottom. A filling need not obey increasing conditions unless these are specified: <standard Young tableaux> use each of $1,\ldots,n$ exactly once and strictly increase along rows and down columns; <Semistandard Young tableaux> permit repetitions and require weakly increasing rows and strictly increasing columns.
In the constructions for a <symmetric group> involving <Young symmetrizers>, <tabloids>, and <polytabloids>, an unqualified numbered <Young tableau> means a bijective filling by $1,\ldots,n$, with no increasing condition imposed. This narrower convention is compatible with the broader use needed for <Semistandard Young tableaux>. Historically, Young's work connected these fillings to <representation theory of the symmetric group>; in particular, the equality between <standard Young tableau> counts and <irreducible representation> degrees over the <complex numbers> predates the modern <hook-length formula>. See https://math.mit.edu/~rstan/algcomb/algcomb.pdf[Stanley's Algebraic Combinatorics, historical notes to Chapter 8].
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