The feasible region for consecutive patterns of permutations is a cycle polytope

TL;DR

This paper characterizes the feasible region for the proportions of consecutive permutation patterns as a cycle polytope of an overlap graph, providing structural insights and independence results between classical and consecutive pattern limits.

Contribution

It establishes that the feasible region is a cycle polytope, computes its geometric properties, and proves the independence of classical and consecutive pattern limits.

Findings

The feasible region is a cycle polytope of an overlap graph.

The dimension, vertices, and faces of the polytope are explicitly computed.

Classical and consecutive pattern limits are shown to be independent.

Abstract

We study proportions of consecutive occurrences of permutations of a given size. Specifically, the limit of such proportions on large permutations forms a region, called \emph{feasible region}. We show that this feasible region is a polytope, more precisely the cycle polytope of a specific graph called \emph{overlap graph}. This allows us to compute the dimension, vertices and faces of the polytope, and to determine the equations that define it. Finally we prove that the limit of classical occurrences and consecutive occurrences are in some sense independent. As a consequence, the scaling limit of a sequence of permutations induces no constraints on the local limit and vice versa.