Semistandard Young tableaux

# 1. Definition

A semistandard (Young) tableau of a partition $\lambda \vdash n$ is map from the cells (also known as boxes) of the Young diagram of $\lambda$, to the natural numbers, such that rows are weakly increasing and the columns are increasing. A semistandard tableau of size $n$ is a semistandard tableau of a partition of size $n$.

All standard (Young) tableaux are also semistandard. The set of semistandard tableaux are in bijection with Gelfand-Tsetlin patterns.

# 3. Properties

• The number of semistandard tableaux of a partition $\lambda$ with all entries in $[r]$ is given by the well known 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 $\lambda$.

• Another similar formula is given by
• $$\prod_{1\leq i < j \leq r} \frac{\lambda_j-\lambda_i + j-i}{j-i}$$ where we implicity pad $\lambda$ with 0 if the index exceeds the number of parts of $\lambda$.