First row of RSK as a minimum-tail table

For a permutation input, the first row of its RSK insertion tableau is the minimum-tail table for increasing subsequences: its $j$th entry is the least attainable final value of a strictly increasing subsequence of length $j$ in the processed prefix. Inserting $z$ replaces the first entry larger than $z$, or appends if none exists. In replacement position $r$, the previous entry in position $r-1$ supplies a subsequence that can be extended by $z$. Shorter minimum tails were already smaller, and a longer subsequence ending at $z$ would require an old tail in position at least $r$ smaller than $z$, contrary to the replacement rule. This proves the invariant inductively. Consequently the first-row length is the longest increasing subsequence length. The row entries need not be a subsequence together: input $2,4,3,1$ gives first row $1,3$, although one occurs after three. They are minima for different lengths, obtained from potentially different subsequences.