---
tag: 287JS93ZDS
kind: definition
name: Young tableau
msc: ["05E10", 05A17]
defines: Q6TH9PH2R7
uses: [096AHMB3P4, 0JV30XN73Z]
sources:
  - url: https://en.wikipedia.org/wiki/Young_tableau
---

## Definition

Let $\lambda = (\lambda_1 \ge \lambda_2 \ge \dots \ge \lambda_\ell > 0)$ be a [[partition|0JV30XN73Z]] of $n$.
Let $D$ be its [[Young diagram|096AHMB3P4]], 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.

1. A *Young tableau* of shape $\lambda$ is a map $T \colon D \to \set{1, 2, 3, \dots}$, called a filling of $D$.
2. $T$ 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.
3. $T$ 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|VCMS1CP91N]].