Standard Young tableau ($\mathrm{SYT}(\lambda )$)

A standard Young tableau fills a Young diagram bijectively with $1,\dots ,n$, increasing along rows and down columns. The cells occupied by the first $r$ entries form a diagram, so a tableau is equivalently a path formed by adjoining one addable node of a Young diagram at each step. The number of standard tableaux of shape $\lambda$ is the dimension of the complex Specht module $S^{\lambda }$.

The content vector of a standard Young tableau $T$ is $(c_T(1),\dots ,c_T(n))$, where $c_T(r)$ is the Content of a Young-diagram cell containing $r$. Such vectors are exactly the integer vectors satisfying: the first coordinate is zero; each subsequent coordinate has an earlier neighbor differing by one; between two occurrences of $a$ both $a-1$ and $a+1$ occur. To reconstruct the tableau, insert each entry in the next available cell on its prescribed diagonal. The neighbor conditions supply its required predecessors, so each insertion is an addable node of a Young diagram.