Shape and pattern containment of separable permutations

TL;DR

This paper investigates how the shape of words under the Robinson-Schensted-Knuth correspondence reflects the containment of separable permutations, providing bounds for supersequence lengths.

Contribution

It establishes that words containing separable permutations have shapes that include the permutation's shape, advancing understanding of pattern containment in combinatorial structures.

Findings

Shapes of words contain the shapes of contained separable permutations

Lower bounds for supersequence lengths of sets with separable permutations

Characterization of pattern containment via shape inclusion

Abstract

Every word has a shape determined by its image under the Robinson-Schensted-Knuth correspondence. We show that when a word w contains a separable (i.e., 3142- and 2413-avoiding) permutation \sigma\ as a pattern, the shape of w contains the shape of \sigma. As an application, we exhibit lower bounds for the lengths of supersequences of sets containing separable permutations.