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 $\mathcal C$ 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.


Introduction
Pattern-avoiding permutations are well-known to show very different behaviour according to the set of patterns they avoid.This makes it extremely difficult to obtain results that are valid for every family of pattern-avoiding permutations.This belief is also confirmed by the available literature, where most of the results are restricted to some specific families of pattern-avoiding permutations.There is one famous exception: Marcus and Tardos [MT04] in 2004 proved the Stanley-Wilf conjecture.Formulated independently by Richard Stanley and Herbert Wilf, it states that for every permutation τ , there is a constant C depending on τ such that the number of permutations of length n which avoid τ is at most C n .
In this paper we introduce the consecutive patterns feasible region for a family C of pattern-avoiding permutations.Several motivations for studying these regions are provided in Section 1.1.We prove a general result -computing their dimension -that holds for instance for all families C avoiding a fixed pattern (see Theorem 1.1).We also study in depth the cases when C is the family of τ -avoiding permutations, with τ of size three or τ a monotone pattern.For these families, we are able to give a complete description of the regions as polytopes (see Theorems 1.14 and 1.16).
1.1.The consecutive patterns feasible regions.The study of limits of (random) pattern-avoiding permutations is a very active field in combinatorics and discrete probability theory.There are two main ways of investigating these limits: • The most classical one is to look at the limits of various statistics for pattern-avoiding permutations.For instance, the limiting distributions of the longest increasing subsequences in uniform pattern-avoiding permutations have been considered in [DHW03, MY19, BBD + 21].Another example is the general problem of studying the limiting distribution of the number of occurrences of a fixed pattern π in a uniform random permutation avoiding a fixed set of patterns when the size tends to infinity (see for instance Janson's papers [Jan17,Jan19,Jan20], where the author studied this problem in the model of uniform permutations avoiding a fixed family of patterns of size three).Many other statistics have been considered: for instance in [BKL + 19] the authors studied the distribution of ascents, descents, peaks, valleys, double ascents and double descents over pattern-avoiding permutations.
• The second way is to look at geometric limits of large pattern-avoiding permutations.Two main notions of convergence for permutations have been defined: a global notion of convergence (called permuton convergence, [HKM + 13]) and a local notion of convergence1 (called Benjamini-Schramm convergence, [Bor20]).For an intuitive explanation of them we refer the reader to [BP20, Section 1.1], where additional references can be found.We just mention here that permuton convergence is equivalent to the convergence of all pattern density statistics (see [BBF + 20, Theorem 2.5]); and Benjamini-Schramm convergence is equivalent to the convergence of all consecutive pattern density statistics (see [Bor20,Theorem 2.19]).The latter is the subject of this paper.In this paper we study the feasible region for consecutive patterns of patternavoiding permutations.This object is strongly connected with both the statistical study and the geometric study of permutations presented above (explanations are given below).We start by defining this region and presenting our main results.
Let S k denote the set of permutations of size k and S the set of all permutations.Many basic concepts on permutations will be recalled in Section 1.5.In this introduction, we use the classical terminology and we briefly introduce essential notation along the way, like c-occ(π, σ) which denotes the proportion of consecutive occurrences of a pattern π in a permutation σ.Given a set of patterns B ⊂ S, we denote by Av n (B) the set of B-avoiding permutations of size n, and by Av(B) := n∈Z ≥0 Av n (B) the set of B-avoiding permutations of arbitrary finite size.We denote by | Av n (B)| the cardinality of Av n (B).
We consider the consecutive patterns feasible region for Av(B), defined by In words, the region P B k is formed by the k!-dimensional vectors v for which there exists a sequence of permutations in Av(B) whose size tends to infinity and whose proportion of consecutive patterns of size k tends to v.For simplicity, whenever B = {τ } we simply write P τ k for P {τ } k (and we use the same convention for related notation).
Our first main result is the following one.We denote by ⊕ the direct sum of two permutations and by the skew sum (definitions are in Section 1.5).
Theorem 1.1.Fix k ∈ Z ≥1 and a set of patterns B ⊂ S such that the family Av(B) is closed either for the ⊕ operation or operation.The feasible region P B k is closed and convex.Moreover, Remark 1.2.We emphasize that the hypothesis in Theorem 1.1 is not superfluous.Indeed, for some sets of patterns B, the region P B k is not convex.For instance, if B = {132, 213, 231, 312}, then Av(B) is the set of monotone permutations.Therefore, the resulting feasible region for consecutive patterns is formed by two distinct points, hence it is not convex.
Remark 1.3.For any fixed pattern τ ∈ S, the family Av(τ ) is either closed for the ⊕ operation (whenever τ is ⊕-indecomposable) or closed for the operation (whenever τ is -indecomposable).
Therefore, by Theorem 1.1, for every pattern τ ∈ S, the region P τ k is closed and convex, and Our theorem is also valid for various families of pattern-avoiding permutations avoiding multiple patterns.For instance all substitution closed-classes satisfy the hypothesis of Theorem 1.1.Substitution closed-classes were first studied by Albert and Atkinson [AA05] and received much attention in various consecutive works.We refer to [BBFS20, Section 2.2] for an introduction to substitution closed-classes.We also remark that Benjamini-Schramm limits of substitution closed-classes were recently investigated in [BBFS20].A well-known example of a substitution closed-class is given by separable permutations.These form the family of permutations avoiding the patterns 2413 and 3142 and they have been consider in several mathematical fields (in enumerative combinatorics and algorithmics [BBL98,AHP15], in real analysis [Ghy17], and in probability theory [SS91, BBF + 18]).
Our second main result shows that the consecutive patterns feasible regions P τ k for τ of size three or τ a monotone pattern is a polytope, and gives a description of the corresponding vertices.Precise statements are given in Theorem 1.14 and Theorem 1.16, after having introduced the required notation.We finally conjecture (see Conjecture 1.13) that, whenever C is closed for the operation ⊕ or , the feasible region is a polytope.
We now comment on the connection between the consecutive patterns feasible regions and the two ways of studying limits of pattern-avoiding permutations mentioned before.For the first one, i.e. the study of various statistics for patternavoiding permutations, the statistic that we consider here is the number of consecutive occurrences of a pattern (see, for instance, the survey of Elizalde [Eli16] for various motivations for studying these patterns).For the second one, i.e. the study of geometric limits, the relation is with Benjamini-Schramm limits (investigated, for instance, in [Bev19,Bor20,BBFS20]).In particular, having a precise description of the regions P B k for all k ∈ Z ≥1 determines all the Benjamini-Schramm limits that can be obtained through sequences of permutations in Av(B).
An orthogonal motivation for investigating the pattern avoiding feasible regions is the problem of packing patterns in pattern avoiding permutations.The classical question of packing patterns in permutations consists in describing the maximum number of occurrences of a pattern π in any permutation of S n (see for instance [Pri97, AAH + 02, Bar04]).More recently, a similar question in the context of pattern-avoiding permutations has been addressed by Pudwell [Pud20].It consists in describing the maximum number of occurrences of a pattern π in any pattern-avoiding permutation.Describing the feasible region for consecutive patterns of pattern-avoiding permutations P B k is a fundamental step for solving the question of finding the asymptotic maximum number of consecutive occurrences of a pattern π ∈ S k in large permutations of Av(B) (indeed the latter problem can be translated into a linear optimization problem in the feasible region P B k ).Additional motivations for studying the regions P B k are the novelties of the results in this paper compared with a previous work [BP20].There, the consecutive patterns feasible region for the set of all permutations S was introduced as: and studied, specifically giving its dimension, establishing that it is a polytope, and describing all its vertices and facets.We refer the reader to [BP20, Section 1.1] for motivations to investigate this region and to [BP20, Section 1.2] for a summary of the related literature.
Remark 1.5.We recall that (classical) patterns feasible regions were also considered in the literature.We refer to [BP20, Section 1.2] for a complete review of the related literature.We also remark that determining the dimension of the (classical) patterns feasible regions for the set of all permutations is still an open problem (see [BP20, Conjecture 1.3]).1.2.Previous results on the standard feasible region for consecutive patterns.Before presenting our additional results on the consecutive patterns feasible regions, we recall two key definitions from [BP20] and review some results presented in that paper.
Definition 1.6.The overlap graph Ov(k) is a directed multigraph with labelled edges, where the vertices are elements of S k−1 and for every π ∈ S k there is an edge labelled by π from the pattern induced by the first k − 1 indices of π to the pattern induced by the last k − 1 indices of π.We define the cycle polytope of G to be the polytope We recall some results from [BP20].We start with the following consequence of [GLS03, Proposition 6].
Our main result in [BP20] is the following one.
Theorem 1.9 (Theorem 1.6 in [BP20]).P k is the cycle polytope of the overlap graph Ov(k).Its dimension is k! − (k − 1)! and its vertices are given by the simple cycles of Ov(k).
An instance of the result above is depicted in Fig. 1. (1,0,0,0,0,0) (0,0,0,0,0,1) Ov(3) respectively.Note that the top vertex (resp.the right-most vertex) of the polytope corresponds to the loop indexed by 123 (resp.321); the other four vertices correspond to the four cycles of length two in Ov(3).We highlight in light-blue one of the six three-dimensional faces of P 3 .This face is a pyramid with a square base.The polytope itself is a four-dimensional pyramid, whose base is the blue face.Theorem 1.9 implies that P 3 is the cycle polytope of Ov(3).
We also recall for later purposes the following construction related to the overlap graph Ov(k).Given a permutation σ ∈ S m , for some m ≥ k, we can associate with it a walk W k (σ) = (e 1 , . . ., e m−k+1 ) in Ov(k) of size m − k + 1, where e i is the edge of Ov(k) labelled by the pattern of σ induced by the indices from i to i + k − 1.The map W k is not injective, but in [BP20] we proved the following.
Lemma 1.10 (Lemma 3.8 in [BP20]).Fix k ∈ Z ≥1 and m ≥ k.The map W k , from the set S m of permutations of size m to the set of walks in Ov(k) of size m − k + 1, is surjective.
This lemma was a key step in the proof of Theorem 1.9.Informally, Ov B (k) arises simply as the restriction of Ov(k) to all the edges and vertices in Av(B).We have the following result, which is proved in Section 2.
Proposition 1.12.Fix k ∈ Z ≥1 .For all sets of patterns B ⊂ S, the feasible region P B k satisfies P B k ⊆ P (Ov B (k)) ⊆ P k .We will show later in Fact 1.15 that sometimes P B k = P (Ov B (k)) even if B = {τ } (see also the bottom part of Fig. 2).Note that this makes the proof of Theorem 1.1 less straightforward.Indeed, only the upper bound dim( | can be deduced from Proposition 1.12 together with Proposition 1.8.As we will see in Section 2, for the complete proof of Theorem 1.1 we use a new approach. Theorem 1.1 states that the regions P B k are convex for every choice of B such that Av(B) is closed either for the ⊕ operation or operation.We further believe that the following stronger result holds.An instance of the result above is depicted on top of Fig. 2.
Despite the description of the region P 312 k is quite simple, for some patterns τ , the precise description of the region P τ k is quite involved, as we will see in the next result.We fix ν n = n • • • 1 for n ∈ Z ≥2 , the decreasing pattern of size n, and an integer k ∈ Z ≥1 .
We start with the following fact (compare this with the bottom part of Fig. 2), which shows that the study of the monotone case deviates significantly from the one in Theorem 1.14.
As a consequence, the feasible region P νn k does not coincide with the cycle polytope of the overlap graph Ov νn (k).In Section 4 we introduce a coloured version of the graph Ov νn (k), denoted COv νn (k), which helps us overcome the problem of the description of the feasible region P νn k through a cycle polytope (see in particular Definition 4.5).
The main result for the monotone patterns case is the following one.
There exists a projection map Π, explicitly described in Eq. (5), such that the consecutive patterns feasible region P νn k is the Π-projection of the cycle polytope of the coloured overlap graph COv νn (k).
That is, ) .An instance of the result stated in Theorem 1.16 is depicted on the bottom part of Fig. 2. We remark that Theorem 1.16 highlights what kind of difficulties can be encountered in proving Conjecture 1.13.1.4.Future projects and open questions.We present here some open questions.
• Theorem 1.14 and Theorem 1.16 give a description of the feasible regions P τ k for all patterns τ of size three.Can we describe the feasible regions P B k for all subsets B ⊆ S 3 ?It is easy to see that P B k ⊆ τ ∈B P τ k , but the reverse inclusion does not hold in general.
• It seems to be the case that the feasible region P B k can be precisely described for other specific sets of patterns B different from the ones already considered in this paper.In particular, we believe that a good choice would be a set of (possibly generalized) patterns B for which the corresponding family Av(B) can be enumerated with generating trees.Indeed, the first author of this article has recently shown in [Bor21] that generating trees behave well in the analysis of consecutive patterns of permutations in these families.We believe that generating trees would be particularly helpful to prove some analogues of Lemma 4.14 -that is the key lemma in the proof of Theorem 1.16 -for other families of permutations.
• The main open question of this article is Conjecture 1.13.1.5.Notation.We present now some notation and simple results that we will use throughout.
Permutations and patterns.We recall that we denoted by S n the set of permutations of size n, and by S the set of all permutations.
Given two permutations, σ ∈ S n for some n ∈ Z ≥1 and π ∈ S k for some k ≤ n, and a set of indices I = {i 1 < . . .< i k }, we say that σ(i 1 ) . . .σ(i k ) is an occurrence of π in σ if pat I (σ) = π (we will also say that π is a pattern of σ).If the indices i 1 , . . ., i k form an interval, then we say that σ(i 1 ) . . .σ(i k ) is a consecutive occurrence of π in σ (we will also say that π is a consecutive pattern of σ).We denote intervals of integers as [n, m] = {n, n+1, . . ., m} for n, m ∈ Z ≥1 with n ≤ m.
We denote by occ(π, σ) the number of occurrences of a pattern π in σ and by c-occ(π, σ) the number of consecutive occurrences of a pattern π in σ.Moreover, we denote by occ(π, σ) (resp.by c-occ(π, σ)) the proportion of occurrences (resp.consecutive occurrences) of a pattern Remark 1.18.The natural choice for the denominator of the expression in the righthand side of the equation above should be n − k + 1 and not n, but we make this choice for later convenience.Moreover, for every fixed k, there is no difference in the asymptotics when n tends to infinity.
For a fixed k ∈ Z ≥1 and a permutation σ ∈ S, we let occ k (σ), c-occ k (σ) ∈ [0, 1] S k be the vectors We say that σ avoids π if σ does not contain any occurrence of π.We point out that the definition of π-avoiding permutations refers to occurrences and not to consecutive occurrences.Given a set of patterns B ⊂ S, we say that σ avoids B if σ avoids π for all π ∈ B. We denote by Av n (B) the set of B-avoiding permutations of size n and by Av(B) := n∈Z ≥1 Av n (B) the set of B-avoiding permutations of arbitrary finite size.The set Av(B) is often called a permutation class.
We also introduce two classical operations on permutations.We denote with ⊕ the direct sum of two permutations, i.e. for τ ∈ S m and σ ∈ S n , and we denote with ⊕ σ the direct sum of copies of σ (we remark that the operation ⊕ is associative).A similar definition holds for the skew sum , (1) We say that a permutation is ⊕-indecomposable (resp.-indecomposable) if it cannot be written as the direct sum (resp.skew-sum) of two non-empty permutations.

Directed graphs.
All graphs, their subgraphs and their subtrees are considered to be directed multigraphs in this paper (and we often refer to them as directed graphs or simply as graphs).In a directed multigraph G = (V (G), E(G)), the set of edges E(G) is a multiset, allowing for loops and parallel edges.An edge e ∈ E(G) is an oriented pair of vertices, (v, u), often denoted by v → u.We write s(e) for the starting vertex v and a(e) for the arrival vertex u.We often consider directed graphs G with labelled edges, and write lb(e) for the label of the edge e ∈ E(G).In a graph with labelled edges we refer to edges by using their labels.Given an edge e ∈ E(G), we denote by C G (e) (for "set of continuations of e") the set of edges e ∈ E(G) such that s(e ) = a(e). A A walk is a path if all the edges are distinct, as well as its vertices, with a possible exception that s(e 1 ) = a(e k ) may happen.A cycle that is a path is called a simple cycle.Given two walks w = (e 1 , . . ., e k ) and w = (e 1 , . . ., e k ) such that a(e k ) = s(e 1 ), we write w • w for their concatenation, i.e. w • w = (e 1 , . . ., e k , e 1 , . . ., e k ).For a walk w, we denote by |w| the number of edges in w.
Given a walk w = (e 1 , . . ., e k ) and an edge e, we denote by n e (w) the number of times the edge e is traversed in w, i.e. n e (w) := |{i ≤ k|e i = e}|.

Topology and dimensions of the consecutive patterns feasible regions
This section is devoted to the proof of Proposition 1.12 and Theorem 1.1.
We start with Proposition 1.12, which states that for all sets of patterns B ⊂ S, the feasible region Recall the map W k associating a walk in Ov(k) to each permutation, defined before Lemma 1.10.
Proof of Proposition 1.12.We start by proving the first inclusion.Consider any point v ∈ P B k , and a corresponding sequence σ ≥0 ∈ Av(B) Z ≥0 such that c-occ k (σ ) → v.Because σ ∈ Av(B), we know that for each , W k (σ ) is a walk in Ov B (k).Using the same method as in the proof that P k ⊆ P (Ov(k)) in [BP20, Theorem 3.12], we can deduce that c-occ k (σ ) converges to a point in P (Ov B (k)).Specifically, recall that W (σ ) is a walk in the graph Ov B (k).We can show that the distance between c-occ k (σ ) and P (Ov B (k)) goes to zero by decomposing W (σ ) into cycles in Ov B (k) and a remaining path with negligible size, and so v ∈ P (Ov B (k)).Because v is generic, it follows that P B k ⊆ P (Ov B (k)).The second inclusion follows from the fact that Ov B (k) is a subgraph of Ov(k) and from Theorem 1.9.
We now turn to the proof of Theorem 1.1.We start by stating a classical consequence of the fact that P B k is a set of limit points.Here we omit the proof: for a similar proof, see [BP20, Lemma 3.1].
Lemma 2.1.Fix k ∈ Z ≥1 .For any set of patterns B ⊆ S, the feasible region P B k is a closed set.
For completeness, we include a simple proof of the statement.Recall that we define c-occ k (σ) := c-occ(π, σ) π∈S k .
Proof.It suffices to show that, for any sequence ( k is convex.Proof.We will present a proof for the case where Av(B) is closed for the ⊕ operation, however the arguments hold equally for the operation.
Since P B k is a closed set (by Lemma 2.1) it is enough to consider rational convex combinations of points in P B k , i.e. it is enough to establish that for all v 1 , v 2 ∈ P B k and all s, t ∈ Z ≥1 , we have that Define t := t • |σ 1 | and s := s • |σ 2 |.We set τ := ⊕ s σ 1 ⊕ ⊕ t σ 2 .We note that for every π ∈ S k , we have where Er ≤ (s + t − 1) • |π|.This error term comes from the number of intervals of size |π| that intersect the boundary of some copies of σ 1 or σ 2 .Hence As tends to infinity, we have This ends the proof of the first part of the statement.
We now prove a result that gives an upper bound on the dimension of P B k .Proposition 2.3.Fix k ∈ Z ≥1 and a set of patterns B ⊂ S such that the class Av(B) is closed for one of the two operations ⊕, .Then the graph Ov B (k) is strongly connected and dim(P (Ov Proof.Consider v 1 , v 2 two vertices of Ov B (k), and assume that Av(B) is closed for ⊕, for simplicity.Then lb(v 1 ) ⊕ lb(v 2 ) is a permutation in Av(B), so W k (lb(v 1 ) ⊕ lb(v 2 )) is a walk in the graph Ov B (k) that connects v 1 to v 2 .We conclude that Ov B (k) is strongly connected.It follows from Proposition 1.8 that We now fix a set of patterns B ⊂ S such that the class Av(B) is closed under the ⊕ operation (the other case is similar).Note that thanks to Propositions 1.12 and 2.3 we have that dim Our strategy to prove Eq. ( 2) is to show that there exists a portion of the polytope k .We start by explicitly describing this portion.
Recall first that from [BP20, Proposition 2.2], if G = (V, E) is a directed multigraph then the vertices of the polytope P (G) are precisely the vectors Consider the vertex e of P (Ov B (k)) corresponding to the loop in Ov B (k) given by the increasing permutation ι k = 1 . . .k (here we are using the fact that B is closed under ⊕).
Let N E k be the set of permutations σ in Av k (B) such that σ(k) = k (N E k stands for not ending with k but also recalls that the permutations in N E k have size k).For π ∈ N E k , we set σ n (π) := ⊕ n (π ⊕ ι k ), for all n ∈ Z >0 and p(π) := lim n→∞ c-occ k (σ n (π)).
We show that the limit is well defined.
Proof.Note that for any See Fig. 3 to clarify the decomposition of c-occ(ρ, The following proposition describes the portion of P (Ov B (k)) that is contained in P B k .Proposition 2.5.The polytope conv ({ p(π) : Proof.Since P B k is convex thanks to Proposition 2.2, it is enough to show that k , for every π ∈ N E k .The first claim follows from the fact that B is closed under ⊕ and therefore the increasing permutations (ι m ) m∈Z>0 avoid B and satisfy c-occ k (ι m ) → e .For the second claim, it is enough to note that σ m (π) = ⊕ m (π⊕ι k ) ∈ Av(B) for all m ∈ Z >0 (where we are using again that B is closed under ⊕) and recall the definition of p(π).
The following proposition guarantees that the polytope has the correct dimension that we need to prove Eq. (2).Proof.We start by defining a partial order ≺ on N E k .For τ 1 , τ 2 ∈ N E k we say that for some m ∈ Z ≥0 .This relation is clearly transitive and reflexive.
To observe that it is also anti-symmetric notice that if τ 1 ≺ τ 2 and τ 2 ≺ τ 1 then there exist two integers m 1 , m 2 ∈ Z ≥0 such that τ 2 = pat [1,k] (ι m1 ⊕ τ 1 ) and Thus, ≺ defines a partial order.Now, consider the collection of linear functionals f ρ ∈ R S k * defined for all ρ ∈ S k by f ρ ( e π ) = δ ρ,π , for all π ∈ S k , where δ denotes the Kronecker delta.We also fix an (arbitrary) extension of the partial order ≺ to a total order and we define the following matrix Because ( p(π)) π∈N E k , e are in P and so also in the affine span of P , we have that the dimension of P is bounded below by the dimension of span{ p(π) − e |π ∈ N E k } .This dimension is bounded below by the rank of A. It suffices then to show that A is upper-triangular with non-zero elements in the diagonal, showing that it is full rank.Indeed, using that First, on the diagonal, we have that f π ( p(π)) = ( p(π)) π = 0 by Lemma 2.4.On the other hand, if ρ, π ∈ N E k , ρ = π, and f ρ ( p(π)) = ( p(π)) ρ is non-zero, then we must have c-occ(ρ, π ⊕ ι k ⊕ pat k−1 (π)) ≥ 1, but because ρ ∈ N E k , it is immediate to see that there exists some m ∈ Z >0 such that ρ = pat [1,k] (ι m ⊕ π) and so π ≺ ρ.
Propositions 2.5 and 2.6 prove Eq. (2) and complete the proof of Theorem 1.1.

The feasible region for 312-avoiding permutations
This section is devoted to the proof of Theorem 1.14.The key step in this proof is to show an analogue of Lemma 1.10 for 312-avoiding permutations.More precisely, we have the following.We first explain how Lemma 3.1 follows from Lemma 3.3 and then we prove the latter.
Proof of Lemma 3.1.In order to prove the claimed surjectivity, given a walk w = (e 1 , . . ., e s ) in Ov 312 (k), we have to exhibit a permutation σ ∈ Av(312) of size s + k − 1 such that W k (σ) = w.We do that by constructing a sequence of s permutations (σ i ) i≤s ∈ (Av(312)) s with size |σ i | = i + k − 1, in such a way that σ is equal to σ s .Moreover, we will have that beg |σi+1|−1 (σ i+1 ) = σ i .
Proof of Lemma 3.3.We have to distinguish two cases.
Case 2: π (k) ∈ [2, k − 1].Consider the point just above (k, π (k)) in the diagram of π and the corresponding point in the last k − 1 points of σ (for an example see the two red points in Fig. 4).Let i be the index in the diagram of σ of the latter point.We claim that σ * σ(i) ∈ Av(312) and end k (σ * σ(i) ) = π .The latter is immediate.It just remains to show that σ := σ * σ(i) ∈ Av(312).
Assume by contradiction that σ contains an occurrence of 312.Since by assumption σ ∈ Av(312) then the value 2 of the occurrence 312 must correspond to the final value σ (|σ |) = σ(i) of σ .Moreover, since π ∈ Av(312), the 312-occurrence cannot occur in the last k elements of σ , that is the 312-occurrence must occur at the values σ (j), σ (r), σ . Therefore, we have two cases: • If r < i then σ (j), σ (r), σ (i) is also an occurrence of 312.A contradiction to the fact that σ ∈ Av(312).
is also an occurrence of 312.A contradiction to the fact that π ∈ Av(312).
This concludes the proof.
Position for the new final value of the permutation σ * σ(i) Building on Proposition 2.2 and Lemma 3.1 we can now prove Theorem 1.14.
Proof of Theorem 1.14.The fact that P 312 k = P (Ov 312 (k)) follows using exactly the same proof of [BP20, Theorem 3.12] replacing Lemma 3.8 and Proposition 3.2 of [BP20] by Lemma 3.1 and Proposition 2.2 of this paper (note that in the proof of [BP20, Theorem 3.12] we also use the fact that the feasible region is closed and this is still true for P 312 k , thanks to Lemma 2.1).For any permutation σ, we define its right-top monotone colouring (simply RITMO colouring henceforth), which we denote as C(σ).This colouring is constructed iteratively, starting with the highest value of the permutation which receives the colour 1 and going down while assigning the lowest possible colour that prevents the occurrence of a monochromatic 21.

The feasible region for monotone-avoiding permutations
If a permutation is coloured with its RITMO colouring, the left-to-right maxima are coloured by 1; removing these left-to-right maxima, the left-to-right maxima of the resulting set of points are coloured by 2, and so on.We suggest to the reader to keep in mind both points of view (the one given in the definition and the one described now) on RITMO colourings.
Example 4.2.In all our examples, we paint in red the values coloured by 1, in blue the ones coloured by two, and in green the ones coloured by three.For instance, the RITMO colouring for permutations 1427536 is given by 1427536.
For the pair (σ, C(σ)) we simply write S(σ).If σ avoids the permutation ν n , it is known that its RITMO colouring is an (n − 1)-colouring (the origins of this result are hard to trace, but it goes back at least to [Gre74] where it is already noted as something that is not hard to prove; see also [Bón12,Chapter 4.3]).
We furthermore allow for taking restrictions of colourings.Given a permutation σ of size k, a colouring c of σ and a subset I = {i 1 , . . ., i j } ⊆ [k], we consider the restriction pat I (σ, c) to be the pair (pat I (σ), c ), where c ( ) = c(i ) for all ∈ [j].
The following definition is fundamental in our results.
Definition 4.3.We say that an m-colouring c of a permutation π ∈ Av(ν n ) of size k is inherited if there is some permutation σ ∈ Av(ν n ) of size ≥ k such that end k (S(σ)) = (π, c).
To sum up, we have introduced three notions of colourings, each more restricted than the previous one.In particular, any RITMO colouring is an inherited colouring, and any inherited colouring is a colouring.
Let C m (π) be the set of all inherited m-colourings of a permutation π ∈ Av(ν n ).We also set C m (k) = {(π, c)|π ∈ Av k (ν n ), c is an inherited m-colouring of π}, that is the set of all inherited m-colourings of permutations of size k.
Example 4.4.Let n = 3.In Table 1 we present all the inherited 2-colourings of permutations of size three.We introduce a key definition for this and the consecutive sections.Table 1.The permutations of size three, and their corresponding inherited 2colourings.Note that all permutations of size four in this table are coloured according to their RITMO colouring.Observe also that the coloured permutation 213 is not inherited.2. Note that in order to obtain a clearer picture we do not draw multiple edges, but we use multiple labels (for example the edge 12 → 21 is labelled with the permutations 231 and 132 and should be thought of as two distinct edges labelled with 231 and 132 respectively).The role of the orange edges will be clarified later.
Definition 4.5.The coloured overlap graph COv νn (k) is defined with the vertex set and the edge set In Fig. 5 we present the coloured overlap graph corresponding to k = 3 and n = 3.
Lemma 4.6.The coloured overlap graph is well-defined, i.e. that for any edge The following simple result is a key step for the proof of the lemma above.
We can now give a more precise formulation of Theorem 1.16.We recall that we denote by δ the Kronecker delta function.
Theorem 4.8.Let Π be the projection map that sends the basis elements (δ (π,c) (x)) x∈Cn−1(k) to (δ π (x)) x∈Av k (νn) , i.e. the map that "forgets" colourings.In this way, the feasible region P νn k is the Π-projection of the cycle polytope of the overlap graph COv νn (k).That is, The feasible region is the projection of the cycle polytope of the coloured overlap graph.To prove Theorem 4.8, we start by recalling that P νn k is a convex set, as established in Proposition 2.2.Thus, in order to prove that P νn k ⊇ Π(P (COv νn (k))) it is enough to show that for any vertex v ∈ P (COv νn (k)) -these vertices are given by the simple cycles of COv νn (k) -its projection Π( v) is in the feasible region.To this end, we construct a walk map CW νn k (see Definition 4.9 below) that transforms a permutation σ ∈ Av(ν n ) into a walk on the graph COv νn (k).Secondly, in order to prove the other inclusion, we see via a factorization theorem that any point in the feasible region results from a sequence of walks in COv νn (k) that can be asymptotically decomposed into simple cycles; so the feasible region must be in the convex hull of the vectors given by simple cycles.
Remark 4.10.Given a permutation σ that avoids ν n , each of the restrictions is an inherited (n − 1)-colouring.The fact that these are (n − 1)-colourings follows because σ avoids ν n , and the fact that these are inherited colourings follows from Observation 4.7 after computations similar to Eq. (4).
Example 4.11.We present the walk CW νn k (σ) corresponding to the permutation σ = 1243756, for k = 3 and n = 3.The RITMO colouring of σ is 1243756, and the corresponding walk is (123,132,213,132,312) .We can see in Fig. 5 this walk highlighted in orange on the coloured overlap graph COv 321 (3).
The following preliminary lemma is fundamental for the proof of Theorem 4.8.Lemma 4.12.There exists a constant C = C(k, n) such that, for any walk w = (e 1 , . . ., e j ) in COv νn (k) there exists a walk w in COv νn (k) of length |w | ≤ C and a permutation σ of size j + k − 1 + |w | that satisfies CW νn k (σ) = w • w.Remark 4.13.Note that, heuristically speaking, Lemma 4.12 states that the map CW νn k is "almost" surjective.This gives an analogue of the result stated in Lemma 3.1 for ν n -avoiding permutations instead of 312-avoiding permutations.
In the same spirit of the proof of Lemma 3.3, in order to prove Lemma 4.12 we need the following result (whose proof is postponed to Section 4.4).Recall the definition of the set C G (e) of continuations of an edge e in a graph G, i.e. the set of edges e ∈ E(G) such that s(e ) = a(e).
Proof of Lemma 4.12.We start by defining the desired constant C = C(k, n).Recall that the edges of the coloured overlap graph COv νn (k) are inherited colourings of permutations.Therefore, for each edge e = (π, c) ∈ E(COv νn (k)) we can choose σ e , one among the smallest ν n -avoiding permutations such that (π, c) = end k (S(σ e )).Define C(k, n) := max e∈E(COv νn (k)) |σ e | + n − k − 1.We claim that this is the desired constant.
We will prove a stronger version of the lemma, by constructing a permutation σ ∈ Av(ν n ) such that C(σ) is a rainbow (n − 1)-colouring and CW νn k (σ) = w • w for some walk w bounded as above.This will be proven by induction on the length of the walk j = |w|.
We first consider the case j = 1.In this case, the walk w = (e 1 ) has a unique edge, and we can select σ = (n − 1) • • • 1 ⊕ σ e1 .In this way, it is clear that C(σ) is a rainbow (n − 1)-colouring, because σ has a monotone decreasing subsequence of size n − 1, while it is clearly ν n -avoiding.Furthermore, because end k (S(σ)) = end k (S(σ e1 )) = e 1 , we have that CW We now consider the case j ≥ 2. Take a walk w = (e 1 , . . ., e j ) in COv νn (k), and consider (by induction hypothesis) the permutation σ such that CW νn k (σ) = w • (e 1 , . . ., e j−1 ) for some walk w of size at most C and such that C(σ) is a rainbow (n − 1)-colouring.
We recover here a proposition from [BP20] that will be important in establishing Theorem 4.8.We can now prove the main result of this section.
Proof of Theorem 4.8.Let σ ∈ Av(ν n ).Let us first establish a formula for c-occ k (σ) with respect to the walk CW νn k (σ) defined in Definition 4.9.Given a permutation ρ with a colouring c we set per(ρ, c) = ρ.Given a walk w in COv νn (k) and a permutation π, we define [π : w] as the number of edges e in w such that per(e) = π.Thus, it easily follows that On the other hand, using [BP20, Proposition 2.2], the vertices of P (COv νn (k)) are given by the simple cycles of the graph COv νn (k).Specifically, the vertices are given by the vectors e C ∈ R Cn−1(k) , for each simple cycle C of COv νn (k), as follows: for each inherited coloured permutation (π, c).In this way, we have that We have that c-occ k (σ ) ).This, together with Proposition 2.2, shows the desired inclusion.
As mentioned before, we explain that Properties 2 and 3 are particular cases of the same general result: consider i < j ∈ [|σ|] with σ(i) < σ(j).Let c = C(σ)(i) and d = C(σ)(j) and assume that c < d.Then there are indices We opt to single out Properties 2 and 3 because these will be enough for our applications.We now introduce a key definition.
We present the following analogue of Lemma 4.16.
We now observe a correspondence between edges of COv νn (k) and active sites of some coloured permutations.
Observation 4.19.Fix an inherited coloured permutation (π 1 , c 1 ) of size k − 1.Then there exists a bijection between the set of edges e ∈ COv νn (k) with s(e) = (π 1 , c 1 ) and the set of active sites (y, f ) of (π 1 , c 1 ).Specifically, this correspondence between edges and active sites is given by the following two maps, which can be easily seen to be inverses of each other: Fix now an inherited coloured permutation (π, c).By definition, there exists some σ 0 that satisfies end |π| (S(σ 0 )) = (π, c).The goal of the next section is to show that, with some mild restrictions on the chosen permutation σ 0 , if (y, f ) is an active site of (π, c) then there exists an index i ∈ We already know that there exists a permutation σ 1 such that end |π|+1 (S(σ 1 )) = (π, c) * (y,f ) ; here we are interested in finding out if σ 1 can arise as an extension of σ 0 .
We introduce two definitions and give some of their simple properties.4.4.The proof of the main lemma.We can now prove Lemma 4.14.We will do this as follows: in order to construct a suitable extension of the permutation σ, we will find a suitable index ι so that σ * ι has the desired coloured pattern at the end.According to Lemma 4.21, fixing the pattern at the end of σ * ι determines an interval of admissible values for ι, and according to Lemma 4.23, fixing the colour of the last entry determines a second interval of admissible values for ι.The key step of the proof is to show that these two intervals have non-trivial intersection.This gives us two intervals that are, by Definition 4.20 and Definition 4.22, nonempty.Our goal is to show that these intervals have a non-trivial intersection, concluding that the desired index ι exists.
Therefore, in both cases we have a contradiction.We now claim that C(σ)(r) < f − 1.Indeed, if C(σ)(r) = f − 1, because p < r we have immediately a contradiction with the maximality of p.Moreover, if C(σ)(r) > f − 1, Property 2 of Lemma 4.16 guarantees that there is some k > p such that σ(k) > σ(r) and C(σ)(k) = f − 1.Again, we have a contradiction with the maximality of p.
Therefore, in both cases we have a contradiction.
Using the two claims above, we can conclude that the intervals in Eqs.(10) and ( 11) have a non-trivial intersection, and therefore the envisaged index ι we were looking for exists.Consequently, we can construct the desired permutation σ * ι .

For
an example with k = 3 see the left-hand side of Fig. 1.Definition 1.7.Let G = (V, E) be a directed multigraph.For each non-empty cycle C in G, define e C ∈ R E such that ( e C ) e := # of occurrences of e in C |C| , for all e ∈ E.

Figure 1 .
Figure1.The overlap graph Ov(3) and the four-dimensional polytope P 3 .The coordinates of the vertices correspond to the patterns (123, 231, 312, 213, 132, 321) respectively.Note that the top vertex (resp.the right-most vertex) of the polytope corresponds to the loop indexed by 123 (resp.321); the other four vertices correspond to the four cycles of length two in Ov(3).We highlight in light-blue one of the six three-dimensional faces of P 3 .This face is a pyramid with a square base.The polytope itself is a four-dimensional pyramid, whose base is the blue face.Theorem 1.9 implies that P 3 is the cycle polytope of Ov(3).

1. 3 .
Additional results on the consecutive patterns feasible regions.We start with a natural generalization of Definition 1.6 to pattern-avoiding permutations.Definition 1.11.Fix a set of patterns B ⊂ S and k ∈ Z ≥1 .The overlap graph Ov B (k) is a directed multigraph with labelled edges, where the vertices are elements of Av k−1 (B) and for every π ∈ Av k (B) there is an edge labelled by π from the pattern induced by the first k − 1 indices of π to the pattern induced by the last k − 1 indices of π.
Conjecture 1.13.Fix k ∈ Z ≥1 and a sets of patterns B ⊂ S such that the family Av(B) is closed either for the ⊕ operation or operation.The feasible region P B k is a polytope.We will prove that Conjecture 1.13 is true when |τ | = 3 or when τ is a monotone pattern, i.e.τ = n • • • 1 or τ = 1 • • • n, for n ∈ Z ≥2 .By symmetry, we only need to study the cases τ = 312 and τ = n • • • 1 for n ∈ Z ≥2 .Indeed, every other permutation arises as compositions of the reverse map (symmetry of the diagram w.r.t. the vertical axis) and the complementation map (symmetry of the diagram w.r.t. the horizontal axis) of the permutations τ = 312 or τ = n • • • 1 for n ∈ Z ≥2 .Beware that the inverse map (symmetry of the diagram w.r.t. the principal diagonal) cannot be used since it does not preserve consecutive pattern occurrences.We conclude this introduction by describing precisely the polytopes P 312 k and P νn k for all ν n = n . . . 1.When τ = 312 the description of the region P 312 k is quite simple; indeed we have the following result.Theorem 1.14.Fix k ∈ Z ≥1 .The feasible region P 312 k is the cycle polytope of the overlap graph Ov 312 (k).

Figure 2 .
Figure 2. We use the same conventions as in Fig. 1 for the coordinates of the vertices of the polytopes.Top: The overlap graph Ov 312 (3) and the three-dimensional polytope P 312 3 .Note that P 312 3 ⊂ P 3 (recall Fig. 1).From Theorem 1.14 we have that P 312 3 is the cycle polytope of Ov 312 (3).Bottom: In light grey the overlap graph Ov 321 (3) and the corresponding three-dimensional cycle polytope P (Ov 321 (3)), which is strictly larger than P 321 3 .The latter feasible region is highlighted in yellow.From Theorem 1.16 we have that P 321 k is the projection (defined precisely in Theorem 4.8) of the cycle polytope of the coloured overlap graph COv 321 (3) (see Definition 4.5 for a precise description).This graph is plotted in the bottom-left side.Note that P 321 3 ⊂ P 3 .
we have that v ∈ P B k .For all s ∈ Z ≥1 , consider a sequence of permutations (σ m s ) m∈Z ≥1 ∈ Av(B) Z ≥1 such that |σ m s | m→∞ − −−− → ∞ and c-occ k (σ m s ) m→∞ − −−− → v s , and some index m(s) of the sequence (σ m s ) m∈Z ≥1 such that for all m ≥ m(s), |σ m s | ≥ s and || c-occ k (σ m s ) − v s || 2 ≤ 1 s .Without loss of generality, assume that m(s) is increasing.For every ∈ Z ≥1 , define σ := σ m( ) .It is easy to show that |σ | →∞ −−−→ ∞ and c-occ k (σ ) →∞ −−−→ v , where we use the fact that v s → v. Furthermore, by assumption we have that σ ∈ Av(B).Therefore v ∈ P B k .Next we prove the convexity of P B k stated in Theorem 1.1.Proposition 2.2.Fix k ∈ Z ≥1 .Consider a set of patterns B ⊂ S such that the class Av(B) is closed for one of the two operations ⊕, .Then, the feasible region P B

Figure 3 .
Figure 3. Any consecutive pattern of size k of σ n (π) will fit in exactly one of the highlighted intervals of indexes.
Lemma 3.1.Fix k ∈ Z ≥1 and m ≥ k.The map W k , from the set Av m (312) of 312-avoiding permutations of size m to the set of walks in Ov 312 (k) of size m−k+1, is surjective.To prove the lemma above we have to introduce the following.Definition 3.2.Given a permutation σ ∈ S n and an integer ∈ [n + 1], we denote by σ * the permutation obtained from σ by appending a new final value equal to and shifting by +1 all the other values larger than or equal to .The proof of Lemma 3.1 is based on the following result.Recall the definition of the set C G (e) of continuations of an edge e in a graph G, i.e. the set of edges e ∈ E(G) such that s(e ) = a(e).Lemma 3.3.Let σ be a permutation in Av(312) such that end k (σ) = π for some π ∈ Av k (312).Let π ∈ Av k (312) such that π ∈ C Ov 312 (k) (π).Then there exists ∈ [|σ| + 1] such that σ * ∈ Av(312) and end k (σ * ) = π .

Figure 4 .
Figure 4.A schema for the proof of Lemma 3.3.

Fix ν n
= n • • • 1, the decreasing pattern of size n ∈ Z ≥1 .In this section we study P νn k and we show that it is related to the cycle polytope of the coloured overlap graph COv νn (k) , presented in Definition 4.5 -this is Theorem 1.16, more precisely restated in Theorem 4.8.4.1.Definitions and combinatorial constructions.We start by introducing colourings of permutations.Definition 4.1 (Colourings and RITMO colourings).Fix an integer m ∈ Z ≥1 .For a permutation σ, an m-colouring of σ is a map c : [|σ|] → [m], which is to be interpreted as a map from the set of indices of σ to [m].An m-colouring c is said to be rainbow when im(c) = [m].

Figure 5 .
Figure5.The coloured overlap graph for k = 3 and n = 3, which also appears in the bottom part of Fig.2.Note that in order to obtain a clearer picture we do not draw multiple edges, but we use multiple labels (for example the edge 12 → 21 is labelled with the permutations 231 and 132 and should be thought of as two distinct edges labelled with 231 and 132 respectively).The role of the orange edges will be clarified later.
Proposition 4.15 (Vertices of the cycle polytope).Let G be a directed graph.The set of vertices of P (G) is precisely { e C | C is a simple cycle of G}.
(7) Π( e C ) = 1 |C| π∈Av k (νn) [π : C] e π .Now let us start by proving the inclusion Π(P (COv νn (k))) ⊆ P νn k .Take a vertex of the polytope P (COv νn (k)), that is a vector e C for some simple cycle C of COv νn (k).Because C is a cycle, we can define the walk C • obtained by concatenating times the cycle C.From Lemma 4.12, there exists a walk w with |w | ≤ C(k, n) and a ν n -avoiding permutation σ of size |w | + |C| + k − 1, such that CW νn k (σ ) = w • C • .The next step is to prove that c-occ k (σ ) →∞ −−−→ Π( e C ).

Figure 6 .
Figure 6.A schema for Lemma 4.16.The left-hand side illustrates Property 1 and the right-hand side illustrates Properties 2 and 3.

Figure 7 .
Figure 7.A schema illustrating Definition 4.20.On top-left the permutation σ = 24351, on top-right the pattern π = 231 induced by the last three indices of σ, and on the bottom the quantities ˜ .