Occurrence graphs of patterns in permutations

TL;DR

This paper introduces occurrence graphs of patterns in permutations, exploring their properties and establishing a link between hereditary graph properties and permutation classes.

Contribution

It defines occurrence graphs for permutation patterns and proves that hereditary graph properties correspond to permutation classes, revealing a new structural connection.

Findings

Every hereditary graph property induces a permutation class

Occurrence graphs encode pattern relationships in permutations

Structural properties of these graphs relate to permutation class characteristics

Abstract

We define the \emph{occurrence graph} $G_p(\pi$) of a pattern $p$ in a permutation $\pi$ as the graph with the occurrences of $p$ in $\pi$ as vertices and edges between the vertices if the occurrences differ by exactly one element. We then study properties of these graphs. The main theorem in this paper is that every \emph{hereditary property} of graphs gives rise to a \emph{permutation class}.