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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 4 Standard tableaux of size 3  
[[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.
Properties

The number of standard tableaux of a partition $\lambda$ is given by the well known hooklength 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.
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 RobinsonSchensted 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]
References
 [Sch61] C. Schensted, Longest increasing and decreasing subsequences, Canad. J. Math. 13(1961), 179191
 [Tym12] Julianna Tymoczko, A simple bijection between standard 3×n tableaux and irreducible webs for sl3, J Algebr Comb (2012) 35:611–632
Sage examples
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.
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