Shuffle Invariance of the Super-RSK Algorithm

As in the $(k,l)$-RSK (Robinson-Schensted-Knuth) of [1], other super-RSK algorithms can be applied to sequences of variables from the set $\{t_1,...,t_k,u_1,...,u_l\}$, where $t_1<...<t_k$, and $u_1<...<u_l$. While the $(k,l)$-RSK of [1] is the case where $t_i<u_j$ for all $i$ and $j$, these other super-RSK's correspond to all the $(\big{(}{{k+l}\atop{k}}\big{)}$ shuffles of the $t$'s and $u$'s satisfying the above restrictions that $t_1<...<t_k$ and $u_1<...<u_l$. We show that the shape of the tableaux produced by any such super-RSK is independent of the particular shuffle of the $t$'s and $u$'s.