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

TL;DR

This paper investigates the shape of sequences generated by irrational rotations after Schensted insertion, revealing a piecewise linear boundary with at most two slopes, described explicitly via continued fractions.

Contribution

It generalizes previous work on monotone subsequences by analyzing the Schensted shape for irrational rotations using lattice monotone sets.

Findings

Boundary of Schensted shape approximated by a two-slope piecewise linear function

Explicit description of the shape in terms of continued fraction expansion

Extension of Boyd and Steele's results to a broader setting

Abstract

For a fixed irrational number $\alpha$ and $n\in \mathbb{N}$, we look at the shape 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 the Schensted shape is approximated by a piecewise linear function with at most two slopes. This piecewise linear function is explicitly described in terms of the continued fraction expansion for $\alpha$. Our results generalize those of Boyd and Steele, who studied longest monotone subsequences. Our proofs are based on a careful analysis of monotone sets in two-dimensional lattices.