The feasible regions for consecutive patterns of pattern-avoiding permutations

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.

arXiv commentNew version including referee's corrections, accepted for publication in Discrete Mathematics