Splittings and Ramsey Properties of Permutation Classes

TL;DR

This paper systematically studies the splittability of permutation classes, revealing that classes avoiding sum-decomposable permutations are splittable, while those avoiding simple permutations are not, and connects these results to graph coloring properties.

Contribution

It provides a comprehensive analysis of splittability in permutation classes, establishing new criteria based on the structure of the avoided permutation.

Findings

Classes Av(q) are splittable if q is sum-decomposable of size ≥4.

Classes Av(q) are unsplittable if q is a simple permutation.

Connections between permutation class splittings and circle graph colorings.

Abstract

We say that a permutation p is 'merged' from permutations q and r, if we can color the elements of p red and blue so that the red elements are order-isomorphic to q and the blue ones to r. A 'permutation class' is a set of permutations closed under taking subpermutations. A permutation class C is 'splittable' if it has two proper subclasses A and B such that every element of C can be obtained by merging an element of A with an element of B. Several recent papers use splittability as a tool in deriving enumerative results for specific permutation classes. The goal of this paper is to study splittability systematically. As our main results, we show that if q is a sum-decomposable permutation of order at least four, then the class Av(q) of all q-avoiding permutations is splittable, while if q is a simple permutation, then Av(q) is unsplittable. We also show that there is a close connection between splittings of certain permutation classes and colorings of circle graphs of bounded clique size. Indeed, our splittability results can be interpreted as a generalization of a theorem of Gy\'arf\'as stating that circle graphs of bounded clique size have bounded chromatic number.