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 $\lambda$.

6. Statistics

• Major index
• Attacking pairs
• Inversions
• size of row and column stabilizer

• St000016: The number of attacking pairs of a tableau.

• St000157: The number of descents of a standard tableau.

• St000017: The number of inversions of a tableau.

• St000009: The charge of the tableau.

• St000169: The cocharge of a standard tableau.

• ST000059: The inversion number of a standard Young tableau as defined by Haglund and Steven....

• St000057: The Shynar inversion number of a standard Young tableau.

• St000075: The orbit size of the tableau under promotion.

7. Maps

• shape
• conjugate
• Schuetzenberger involution
• promotion