Enumerating cycles in the graph of overlapping permutations

TL;DR

This paper studies the structure of the graph of overlapping permutations, focusing on the existence, enumeration, and properties of cycles within this graph, which is analogous to the De Bruijn graph.

Contribution

It provides new insights into the cycle structure of the overlapping permutations graph, including counting 2-cycles and characterizing vertices in closed walks.

Findings

Determined conditions for the existence of directed cycles.

Counted the number of 2-cycles in the graph.

Enumerated vertices involved in closed walks and longer cycles.

Abstract

The graph of overlapping permutations is a directed graph that is an analogue to the De Bruijn graph. It consists of vertices that are permutations of length $n$ and edges that are permutations of length $n+1$ in which an edge $a_1\cdots a_{n+1}$ would connect the standardization of $a_1\cdots a_n$ to the standardization of $a_2\cdots a_{n+1}$. We examine properties of this graph to determine where directed cycles can exist, to count the number of directed $2$-cycles within the graph, and to enumerate the vertices that are contained within closed walks and directed cycles of more general lengths.