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# 1. Definition & Example

• A semistandard (Young) tableau of a partition $\lambda \vdash n$ is a map from the cells of the Young diagram of $\lambda$ to $\{1,2,\dots,r\}$ for some $r$, such that rows are weakly and columns are strictly increasing.

• The set of semistandard tableaux of shape $\lambda \vdash n$ with maximal possible entry $r$ is denoted by $\mathcal{SSYT}_{\lambda,r}$, and $$\mathcal{SSYT}_{n,r} = \bigcup_{\lambda \vdash n}\mathcal{SSYT}_{\lambda,r}$$ denotes the set of semistandard tableaux of size $n$.

 The four semistandard tableaux of size 2 and maximal entry 2 [[1,1]] [[1,2]] [[2,2]] [[1],[2]]
• Semistandard tableaux are graphically represented by filling the cells of the Young diagram in English notation.

# 2. FindStat representation and coverage

• A semistandard tableau is uniquely represented as a list of lists giving the fillings of the cells row by row.
• Semistandard tableaux are graded by $n$ where $n$ is both the size and the maximal possible entry.

• The database contains all semistandard tableaux of size/maximal possible entry at most 6.

## 3.1. Properties

• The number of semistandard tableaux of a partition $\lambda$ with all entries in $\{1,\ldots,r\}$ is given by the hook-content formula $$\big|\mathcal{SST}_\lambda\big| = \prod_{(i,j)}\frac{r+i-j}{hook(i,j)},$$ where the product ranges over all boxes $(i,j)$ in the Young diagram of $\lambda$, and where $hook(i,j)$ is the hook length of the box$(i,j)$ in $\lambda$.

## 3.2. Remarks

• Semistandard tableaux are in bijection with Gelfand-Tsetlin patterns.

• Skew Tableaux come from taking the set difference of two partitions of different sizes. If the resulting boxes are filled with numbers such that the columns are increasing and the rows are weakly increasing, then it is called a semistandard skew tableaux.

• The number of semistandard tableaux of shape $\lambda$ and content $\mu$ is equal to the Kostka number $K_{\lambda\mu}$. The hook length formula is a special case when $\mu = (1,1,1,...,1)$, since that corresponds to a standard tableau. However, there is no general formula for the Kostka numbers.