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arXiv · 2107.11515

Monotone subsets in lattices and the Schensted shape of a Sós permutation

Karl Liechty·T. Kyle Petersen
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Abstract

For a fixed irrational number $α$ and $n\in \mathbb{N}$, we look at the shape of the sequence $(f(1),\ldots,f(n))$ after Schensted insertion, where $f(i) = αi \mod 1$. 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 $α$. 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.

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Karl Liechty, T. Kyle Petersen. 2021-07-24. Monotone subsets in lattices and the Schensted shape of a Sós permutation. https://arxiv.org/abs/2107.11515

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