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

Jacopo Borga,Raul Penaguiao

doi:10.5802/alco.135

Abstract

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

Highlights

Despite this article not containing any probabilistic result, we introduce here some motivations that come from the study of random permutations
We present the definition of shift-invariant permutation in Definition 3.3, and we prove the equality in Eq (2) in Proposition 3.4
In this paper we study the feasible region Pk of limiting points for consecutive pattern occurrences

Summary

Introduction

A natural question, motivated by the theorems above, is the following: given a finite family of patterns A ⊆ S and a vector (∆π)π∈A ∈ [0, 1]A, or (Γπ)π∈A ∈ [0, 1]A, does there exist a sequence of permutations (σn)n∈N such that |σn| → ∞ and occ(π, σn) → ∆π, for all π ∈ A, or c-occ(π, σn) → Γπ, for all π ∈ A ?. The feasible region clPk was first studied in [17] for some particular families of patterns instead of the whole Sk. More precisely, given a list of finite sets of permutations In order to find the dimension, the vertices and the equations describing Pk, we first recover and prove general results for cycle polytopes of directed multigraphs (see Section 1.5). In this paper we study the feasible region Pk of limiting points for consecutive pattern occurrences.

The cycle polytope of a graph

The feasible region Pk is a cycle polytope

Mixing classical patterns and consecutive patterns