Occurrence graphs of patterns in permutations

Abstract

We define the occurrence graph $G_p\left(\pi \right)$ of a pattern $p$ in a permutation $\pi$ as the graph whose vertices are the occurrences of $p$ in $\pi$, with 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 hereditary property of graphs gives rise to a permutation class.