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1. Definition & Example
A standard (Young) tableau of a partition $\lambda \vdash n $ is a bijection between $\{1,2,\dots, n\}$ and the cells (also known as boxes) of the Young diagram of $\lambda$, such that rows and columns are increasing.
The set of standard tableaux of size $n$ is denoted by $\mathcal{SYT}_n$ where the size is given by the size of the underlying partition $\lambda \vdash n$.
The four standard tableaux of size 3 |
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[[1,2,3]] |
[[1,3],[2]] |
[[1,2],[3]] |
[[1],[2],[3]] |
Standard tableaux are graphically represented by filling the cells of the Young diagram in English notation.
2. Properties
The number of standard tableaux of a partition $\lambda$ is given by the well known hook-length formula $$ \big|\mathcal{SYT}_\lambda\big| = \frac{n!}{\prod hook(B)},$$ 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 of size $n$ cells is counted by the involution numbers, with the first few terms (beginning with $n=0$) being 1, 1, 2, 4, 10, 26.
3. 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]
4. References
[Tym12] Julianna Tymoczko, A simple bijection between standard 3×n tableaux and irreducible webs for sl3, J Algebr Comb (2012) 35:611–632.
5. Sage examples
6. Technical information for database usage
- A standard tableau is uniquely represented as a list of lists giving the fillings of the cells row by row.
- Standard tableaux are graded by the size.
- The database contains all standard tableaux of size at most 8.