Standard Young tableaux

# 1. Definition

A standard (Young) tableau of a partition $\lambda \vdash n$ is a bijection between the positive integers $\{1,2,\dots, n\}$ and the cells (also known as boxes) of the Young diagram of $\lambda$, such that rows and columns are increasing. (English convention). A standard tableau of size $n$ is a standard tableau of a partition of size $n$.

# 2. Notation

By $\mathcal{ST}_n$, we denote the collection of all standard tableaux of $n$.

# 3. Examples

• $\mathcal{ST}_1 = \{ [[1]] \}$

• $\mathcal{ST}_2 = \{ [[1,2]],[[1],[2]]\}$

• $\mathcal{ST}_3 = \{ [[1,2,3]],[[1,2],[3]],[[1,3],[2]],[[1],[2],[3]]\}$

Here is another example from $\mathcal{ST}_9$.

$\begin{matrix}1 & 4 & 8\\2 & 5 & 9\\3 & 6\\7\\\end{matrix}$ which in Sage input form is $[[1,4,8], [2,5,9],[3,6],[7]]$.

# 4. Properties

• The number of standard tableaux of a partition $\lambda$ is given by the well known hook-length formula $$\big|\mathcal{ST}_\lambda\big| = \frac{n!}{\prod hook(B)},$$ where the product ranges over all boxes $B$ in the Young diagram of $\lambda$, and where $hook(B)$ is the hook length of $B$ in $\lambda$.

# 7. Maps

• shape
• conjugate
• Schuetzenberger involution
• promotion