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For a fixed irrational number $\alpha$ and $n\in \mathbb{N}$, we look at theshape of the sequence $(f(1),\ldots,f(n))$ after Schensted insertion, where$f(i) = \alpha i \mod 1$. Our primary result is that the boundary of theSchensted shape is approximated by a piecewise linear function with at most twoslopes. This piecewise linear function is explicitly described in terms of thecontinued fraction expansion for $\alpha$. Our results generalize those of Boydand Steele, who studied longest monotone subsequences. Our proofs are based ona careful analysis of monotone sets in two-dimensional lattices.

Monotone subsets in lattices and the Schensted shape of a S\'os permutation