The feasible regions for consecutive patterns of pattern-avoiding permutations

TL;DR

This paper investigates the geometric structure of the set of possible limiting frequencies of consecutive patterns in large pattern-avoiding permutations, establishing convexity and conjecturing polytopal nature, with full descriptions in specific cases.

Contribution

It characterizes the feasible regions for consecutive pattern frequencies in pattern-avoiding permutations, proving convexity and providing a full description for certain classes, including cycle polytope vertices.

Findings

Feasible regions are always convex.

These regions are polytopes in specific cases.

Vertices are described via cycle polytopes.

Abstract

We study the feasible region for consecutive patterns of pattern-avoiding permutations. More precisely, given a family $\mathcal C$ of permutations avoiding a fixed set of patterns, we consider the limit of proportions of consecutive patterns on large permutations of $\mathcal C$. These limits form a region, which we call the consecutive patterns feasible region for $\mathcal C$. We determine the dimension of the consecutive patterns feasible region for all families $\mathcal C$ closed either for the direct sum or the skew sum. These families include for instance the ones avoiding a single pattern and all substitution-closed classes. We further show that these regions are always convex and we conjecture that they are always polytopes. We prove this conjecture when $\mathcal C$ is the family of $\tau$-avoiding permutations, with either $\tau$ of size three or $\tau$ a monotone pattern. Furthermore, in these cases we give a full description of the vertices of these polytopes via cycle polytopes. Along the way, we discuss connections of this work with the problem of packing patterns in pattern-avoiding permutations and to the study of local limits for pattern-avoiding permutations.