Young tableau

Builds on Young diagram Integer partition

Definition

Let $\lambda = (\lambda_1 \ge \lambda_2 \ge \dots \ge \lambda_\ell > 0)$ be a partition of $n$ . Let $D$ be its Young diagram, the set of cells $(i, j)$ with $1 \le i \le \ell$ and $1 \le j \le \lambda_i$ , where $i$ is the row and $j$ the column.

A Young tableau of shape $\lambda$ is a map $T \colon D \to \set{1, 2, 3, \dots}$ , called a filling of $D$ .
$T$ is semistandard if $T(i, j) \le T(i, j+1)$ and $T(i, j) < T(i+1, j)$ whenever both cells lie in $D$ . So entries weakly increase along rows and strictly increase down columns.
$T$ is standard if it is semistandard and $T$ is a bijection from $D$ onto $\set{1, \dots, n}$ .

Remarks

Example: for $\lambda = (2, 2)$ , the filling with rows $1\,2$ and $3\,4$ is standard. Each row increases ( $1 < 2$ , $3 < 4$ ), each column increases ( $1 < 3$ , $2 < 4$ ), and each of $1, 2, 3, 4$ appears once. The filling with rows $1\,1$ and $2\,3$ is semistandard but not standard, because $1$ appears twice.

Non-example: the filling of shape $(2, 1)$ with first row $2\,1$ and second row $3$ is not semistandard, because the first row decreases.

Many authors say "tableau" for a semistandard tableau. The content of a semistandard tableau $T$ is the sequence $(\mu_1, \mu_2, \dots)$ , where $\mu_k$ is the number of entries equal to $k$ . The number $f^\lambda$ of standard tableaux of shape $\lambda$ is given by the hook length formula.

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Integer partition concept
Young diagram concept
Young tableau definition

Tag	`287JS93ZDS`
URI	`m2://287JS93ZDS`
Kind	definition
Writing	W0 stub
MSC	05E10, 05A17
Defines	Young tableau
Uses	Young diagram, Integer partition
Changed	2 November 2026

Young tableau

Definition

Remarks

Reading path

Further reading

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