Content vector of a standard Young tableau ($c_T$)

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.

The axial distance from label $j$ to label $i$ in a standard Young tableau is the difference of their Young-diagram cell contents. This sign convention makes $c_T(r+1)-c_T(r)$ equal to $+1$ for consecutive entries in a row and $-1$ in a column. The reciprocal distance supplies the diagonal coefficient of Young orthogonal form.