Robinson–Schensted correspondence ($\pi ⟷(P,Q)$)

Insert the successive letters of a permutation by row insertion to obtain an insertion tableau $P$, recording the insertion times in the new boxes of $Q$. This is a bijection between permutations and pairs of standard Young tableaux of a common shape. Reverse insertion removes the box marked by the latest recording time. The first row of $P$ has length equal to the longest increasing subsequence.

For a finite $0$-$1$ matrix, list its occupied pairs by increasing top entry and decreasing bottom entry within a top-entry tie. Perform ordinary semistandard row insertion on the bottom entries and record top entries in the new cells. The result is a bijection with same-shape tableaux $T,U$ for which $T$ and $U^t$ are Semistandard Young tableaux. The types of $T$ and $U$ are respectively the column-sum and row-sum vectors of the matrix.