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 $\lambda$. The hook length of $B$ is the number of boxes in the same row to the right of $B$, plus the number of boxes in the same column below $B$ plus the box itself.The number of standard tableaux with $n$ cells is counted by the

**involution numbers**, with the first few terms (beginning with $n=0$) being 1, 1, 2, 4, 10, 26.

# 5. Remarks

The number of standard tableaux of shape $(n,n)$ is counted by the $n^{th}$ Catalan number, $\operatorname{C}_n = \frac{1}{n+1} \binom{2n}{n}$.

There is a bijection between permutations $ \pi \in \mathbf{S}_n $ and pairs $(P,Q)$ of Standard Tableaux of shape $\lambda \vdash n $, known as the Robinson–Schensted correspondence. Furthermore, the length of the longest increasing subsequence of $\pi$ is equal to the length of the first row of $P$ (or $Q$), and the length of the longest decreasing subsequence of $\pi$ is equal to the height of the first column of $P$. [Sch61]

There is a bijection between Standard Young Tableaux of shape $(3×n)$ and irreducible webs for ${sl_3}$ whose boundary vertices are all sources [Tym12]

# 6. Statistics

We have the following **13 statistics** in the database:

# 7. Maps

We have the following **6 maps** in the database:

# 8. References

*Longest increasing and decreasing subsequences*, Canad. J. Math.

**13**(1961), 179-191.

[Tym12] Julianna Tymoczko, *A simple bijection between standard 3×n tableaux and irreducible webs for sl3*, J Algebr Comb (2012) **35**:611–632.

# 9. Sage examples