A simple proof of a theorem of Schmerl and Trotter for permutations

TL;DR

This paper provides an elementary proof of Schmerl and Trotter's theorem for permutations, showing that every non-parallel alternation simple permutation contains a smaller simple permutation.

Contribution

It offers a simplified, elementary proof of a key theorem in permutation theory, making the result more accessible.

Findings

Every simple permutation not a parallel alternation contains a simpler simple permutation.

The proof simplifies understanding of the structure of simple permutations.

Supports further research in permutation pattern theory.

Abstract

When specialized to the context of permutations, Schmerl and Trotter's Theorem states that every simple permutation which is not a parallel alternation contains a simple permutation with one fewer entry. We give an elementary proof of this result.