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 12 statistics in the database:

The charge of a standard tableau.
The number of attacking pairs of a standard tableau.
The number of inversions of a standard tableau.
The Shynar inversion number of a standard tableau.
The inversion number of a standard tableau as defined by Haglund and Stevens.
The orbit size of a standard tableau under promotion.
The number of descents of a standard tableau.
The cocharge of a standard tableau.
The major index of a standard tableau.
The leg major index of a standard tableau.
The number of ascents of a standard tableau.
Eigenvalues of the random-to-random operator acting on a simple module.

# 7. Maps

We have the following 6 maps in the database:

to Gelfand-Tsetlin pattern
shape
conjugate
Schuetzenberger involution
catabolism

# 8. References

[Sch61]   C. Schensted, 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

StandardTableaux (last edited 2015-12-18 05:53:12 by ZachariahNeville)