[3] The feasible regions for consecutive patterns of pattern-avoiding permutations (with Raul Penaguiao)

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 study the limit of proportions of consecutive patterns on large permutations of \(\mathcal C\) . These limits form a region, which we call the pattern-avoiding feasible region for \(\mathcal C\) . We show that, when \(\mathcal C\) is the family of \(\tau\) -avoiding permutations, with either \(\tau\) of size three or \(\tau\) a monotone pattern, the pattern-avoiding feasible region for \(\mathcal C\) is a polytope. We also determine its dimension using a new tool for the monotone pattern case whereby we are able to compute the dimension of the image of a polytope after a projection.

We further show some general results for the pattern-avoiding feasible region for any family \(\mathcal C\) of permutations avoiding a fixed set of patterns, and we conjecture a general formula for its dimension.

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.