<> Definition & Example ==================== - A **standard (Young) tableau** of a [partition](/IntegerPartitions) $\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 $\lambda \vdash n$. <> - Standard tableaux are graphically represented by filling the cells of the Young diagram in **English notation**. Properties ========== - The number of standard tableaux of a [partition](/IntegerPartitions) $\lambda$ 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](https://oeis.org/A000085). Remarks ======= - The number of standard tableaux of shape $(n,n)$ is counted by the $n^{th}$ [Catalan number](http://en.wikipedia.org/wiki/Catalan_number), $\operatorname{C}_n = \frac{1}{n+1} \binom{2n}{n}$. - There is a bijection between [permutations](/Permutations) $ \pi \in \mathbf{S}_n $ and pairs $(P,Q)$ of Standard Tableaux of shape $\lambda \vdash n $, known as the [Robinson--Schensted correspondence.](https://en.wikipedia.org/wiki/Robinson%E2%80%93Schensted_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 $P$. <> - There is a bijection between Standard Young Tableaux of shape $(3×n)$ and irreducible webs for ${sl_3}$ whose boundary vertices are all sources <> References ========== - [Wikipedia](http://en.wikipedia.org/wiki/Young_tableau) <> Sage examples ============= {{{#!sagecell for n in [2,3,4]: Ps = StandardTableaux(n) print n, Ps.cardinality() for P in StandardTableaux(3): print P }}} 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.