The feasible regions for consecutive patterns of pattern-avoiding permutations

We study the feasible region for consecutive patterns of pattern-avoiding permutations. More precisely, given a family C of permutations avoiding a fixed set of patterns, we consider the limit of proportions of consecutive patterns on large permutations of C. These limits form a region, which we call the consecutive patterns feasible region for C. We determine the dimension of the consecutive patterns feasible region for all families 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 C is the family of t-avoiding permutations, with either t of size three or t 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.(c) 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).