Given a semistandard Young tableau $T$ of shape $\lambda$, let $c_i(T)$ is the number of $i$'s that appear in $T$. We can define a partial order on the set of all semistandard Young tableau of some fixed shape $\lambda$ by setting $T > U$ if and only if $c_i(T) > c_i(U)$ where $i$ is the minimum $i$ in which $c_i(T)$ and $c_i(U)$ differ. For example, $$\begin{array} & 1 & 1 & 1 \\ 2 & 2 \\ 3 \end{array} \quad > \quad \begin{array} & 1 & 1 & 1 \\ 2 & 3 \\ 3 \end{array}$$ since the first tableau has the the same number of $1$'s as the second but has more $2$'s than the second tableau (note that the number of $3$'s does not matter).
My question: Does this partial order have a name? If so, can anyone point me in the direction of papers/textbooks that use it?
