Consider $p,q \in SSYT(\lambda)$ such that $l(p) = l(q) \leq 2$. I was hoping to find the image under the rsk algorithm of $(p,q)$? In other words the two lines-arrays (or generalized permutations) under the rsk bijetion.
What is known is that if I restrict the algorithm to $SYT$ if find a real permutation so I can split cases on $l(\lambda) = 1$ and $l(\lambda) = 2$. The first should be only $1_{S_n}$ since it determines the SYT with strictly increasing numbers.
This seems really far from the general case I want to solve. I think it's relatively easy to see that a necessary condition is that we have the first jump only if we approached to the second row of the matrix associated to the $SSYT$ (i.e the preimage under the RSK algorithm).
What I don't see is how to derive a general formula to bound the "jump" to two, depending on the rows or columns of the matrix associated to the $SSYT$ since the matrix is random in general. What I do see is that an element in position $(i,j)$ can only make jump elements $(i',j')$ with $j'>i'$. Is this true?
Any help or solution in order to prove the general case would be appreciated.