The feasible region for consecutive patterns of permutations is a cycle polytope
Jacopo Borga, Raul Penaguiao
备注：
New version including referee's corrections. This is an extended abstract of arXiv:1910.02233 for FPSAC 2020 (accepted for publication in a proceedings volume of Séminaire Lotharingien Combinatoire)
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.
We study proportions of consecutive occurrences of permutations of a given size. Specifically, the limit of such proportions on large permutations forms a region, called \emph{feasible region}. We show that this feasible region is a polytope, more precisely the cycle polytope of a specific graph called \emph{overlap graph}. This allows us to compute the dimension, vertices and faces of the polytope, and to determine the equations that define it. Finally we prove that the limit of classical occurrences and consecutive occurrences are in some sense independent. As a consequence, the scaling limi…
Let $π_n$ be a uniformly chosen random permutation on $[n]$. Using an analysis of the probability that two overlapping consecutive $k$-permutations are order isomorphic, we show that the expected number of distinct consecutive patterns in $π_n$ is $\frac{n^2}{2}(1-o(1))$. This exhibits the fact that random permutations pack consecutive patterns near-perfectly.
We consider uniform random permutations drawn from a family enumerated through generating trees. We develop a new general technique to establish a central limit theorem for the number of consecutive occurrences of a fixed pattern in such permutations. We propose a technique to sample uniform permutations in such families as conditioned random colored walks. Building on that, we derive the behavior of the consecutive patterns in random permutations studying properties of the consecutive increments in the corresponding random walks. The method applies to families of permutations with a one-dimen…
We introduce several classes of polytopes contained in $[0,1]^n$ and cut out by inequalities involving sums of consecutive coordinates. We show that the normalized volumes of these polytopes enumerate circular extensions of certain partial cyclic orders. Among other things this gives a new point of view on a question popularized by Stanley. We also provide a combinatorial interpretation of the Ehrhart $h^*$-polynomials of some of these polytopes in terms of descents of total cyclic orders. The Euler numbers, the Eulerian numbers and the Narayana numbers appear as special cases.
In [Baumeister, H., Nill, Paffenholz, On permutation polytopes, Adv. Math. 222 (2009), 431-452 / arXiv:0709.1615] we conjectured a characterization of subgroups H of a permutation group G so that, on the level of permutation polytopes, P(H) is a face of P(G). Here we present the embarrassingly simple proof of this conjecture.
This paper focuses on determining the volumes of permutation polytopes associated to cyclic groups, dihedral groups, groups of automorphisms of tree graphs, and Frobenius groups. We do this through the use of triangulations and the calculation of Ehrhart polynomials. We also present results on the theta body hierarchy of various permutation polytopes.
A permutation polytope is the convex hull of a group of permutation matrices. In this paper we investigate the combinatorics of permutation polytopes and their faces. As applications we completely classify permutation polytopes in dimensions 2,3,4, and the corresponding permutation groups up to a suitable notion of equivalence. We also provide a list of combinatorial types of possibly occuring faces of permutation polytopes up to dimension four.
We introduce Lipschitz functions on a finite partially ordered set $P$ and study the associated Lipschitz polytope $L(P)$. The geometry of $L(P)$ can be described in terms of descent-compatible permutations and permutation statistics that generalize descents and big ascents. For ranked posets, Lipschitz polytopes are centrally-symmetric and Gorenstein, which implies symmetry and unimodality of the statistics. Finally, we define $(P,k)$-hypersimplices as generalizations of classical hypersimplices and give combinatorial interpretations of their volumes and $h^*$-vectors.
Let $S_n$ denote the set of permutations of $[n]:=\{1,\cdots, n\}$, and denote a permutation $σ\in S_n$ by $σ=σ_1σ_2\cdots σ_n$. For $l\ge2$ an integer, let $A^{(n)}_{l;k}\subset S_n$ denote the event that the set of $l$ consecutive numbers $\{k, k+1,\cdots, k+l-1\}$ appears in a set of consecutive positions: $\{k,k+1,\cdots, k+l-1\}=\{σ_a,σ_{a+1},\cdots, σ_{a+l-1}\}$, for some $a$. For $τ\in S_m$, let $S_n(τ)$ denote the set of $τ$-avoiding permutations in $S_n$, and let $P_n^{\text{av}(τ)}$ denote the uniform probability measure on $S_n(τ)$. Also, let $S_n^{\text{sep}}$ denote the set of sep…
We give an explicit formula for the number of permutations avoiding cyclically a consecutive pattern in terms of the spectrum of the associated operator of the consecutive pattern. As an example, the number of cyclically consecutive $123$-avoiding permutations in ${\mathfrak S}_{n}$ is given by $n!$ times the convergent series ${\displaystyle \sum_{k=-\infty}^{\infty} \left(\frac{\sqrt{3}}{2π(k+1/3)}\right)^{n}}$ for $n \geq 2$.
We investigate the combinatorics and geometry of permutation polytopes associated to cyclic permutation groups, i.e., the convex hulls of cyclic groups of permutation matrices. We give formulas for their dimension and vertex degree. In the situation that the generator of the group consists of at most two orbits, we can give a complete combinatorial description of the associated permutation polytope. In the case of three orbits the facet structure is already quite complex. For a large class of examples we show that there exist exponentially many facets.
In their study of cyclic pattern containment, Domagalski et al. conjecture differential equations for the generating functions of circular permutations avoiding consecutive patterns of length 3. In this note, we prove and significantly generalize these conjectures. We show that, for every consecutive pattern $σ$ beginning with 1, the bivariate generating function counting occurrences of $σ$ in circular permutations can be obtained from the generating function counting occurrences of $σ$ in (linear) permutations. This includes all the patterns for which the latter generating function is known.
Symmetry plays a fundamental role in design of experiments. In particular, symmetries of factorial designs that preserve their statistical properties are exploited to find designs with the best statistical properties. By using a result proved by Rosenberg [6], the concept of the LP relaxation orthogonal array polytope is developed and studied. A complete characterization of the permutation symmetry group of this polytope is made. Also, this characterization is verified computationally for many cases. Finally, a proof is provided.
We define and study a new family of polytopes which are formed as convex hulls of partial alternating sign matrices. We determine the inequality descriptions, number of facets, and face lattices of these polytopes. We also study partial permutohedra that we show arise naturally as projections of these polytopes. We enumerate facets and also characterize the face lattices of partial permutohedra in terms of chains in the Boolean lattice. Finally, we have a result and a conjecture on the volume of partial permutohedra when one parameter is fixed to be two.
In 2012, Sagan and Savage introduced the notion of $st$-Wilf equivalence for a statistic $st$ and for sets of permutations that avoid particular permutation patterns which can be extended to generalized permutation patterns. In this paper we consider $inv$-Wilf equivalence on sets of two or more consecutive permutation patterns. We say that two sets of generalized permutation patterns $Π$ and $Π'$ are $inv$-Wilf equivalent if the generating function for the inversion statistic on the permutations that simultaneously avoid all elements of $Π$ is equal to the generating function for the inversio…
In this paper, we introduce a new method for computing generating functions with respect to the number of descents and left-to-right minima over the set of permutations which have no consecutive occurrences of a pattern that starts with 1.
A consecutive pattern in a permutation $π$ is another permutation $σ$ determined by the relative order of a subsequence of contiguous entries of $π$. Traditional notions such as descents, runs and peaks can be viewed as particular examples of consecutive patterns in permutations, but the systematic study of these patterns has flourished in the last 15 years, during which a variety of different techniques have been used. We survey some interesting developments in the subject, focusing on exact and asymptotic enumeration results, the classification of consecutive patterns into equivalence classe…
We clarify the notion of effective equivalence and characterize geometrically the effectively equivalent permutation groups. In particular, we present examples showing that the latter do not correspond to affinely equivalent polytopes thereby answering Question 2.12 of [BHNP09]. We apply our characterization to our examples and formulate several questions.
We present a new approach to the problem of enumerating permutations of length n that avoid a fixed consecutive pattern of length m. We use this idea to give explicit upper and lower bounds on the number of permutations avoiding a pattern of length m. As a corollary, we obtain a simple proof of the CMP conjecture, regarding the most avoided pattern, recently shown by Elizalde. Finally, we also show that most of the patterns behave similar to the least avoided one.
We exploit Krattenthaler's bijection between the set $S_n(3\textrm{-}1\textrm{-}2)$ of permutations in $S_n$ avoiding the classical pattern $3\textrm{-}1\textrm{-}2$ and Dyck $n$-paths to study the distribution of every consecutive pattern of length 3 on the set $S_n(3\textrm{-}1\textrm{-}2)$. We show that these consecutive patterns split into 3 equidistribution classes, by means of an involution on Dyck paths due to E.Deutsch. In addition, we state equidistribution theorems concerning triplets of statistics relative to the occurrences of the consecutive patterns of length 3 in a permutation.
In recent years, there has been increasing interest in consecutive pattern avoidance in permutations. In this paper, we introduce two approaches to counting permutations that avoid a set of prescribed patterns consecutively. These algoritms have been implemented in the accompanying Maple package CAV, which can be downloaded from the author's website. As a byproduct of the first algorithm, we have a theorem giving a sufficient condition for when two pattern sets are strongly (consecutively) Wilf-Equivalent. For the implementation of the second algorithm, we define the cluster tail generating fu…
Each group G of nxn permutation matrices has a corresponding permutation polytope, P(G):=conv(G) in R^{nxn}. We relate the structure of P(G) to the transitivity of G. In particular, we show that if G has t nontrivial orbits, then min{2t,floor(n/2)} is a sharp upper bound on the diameter of the graph of P(G); so if G is transitive, the diameter is at most 2. We also show that P(G) achieves its maximal dimension of (n-1)^2 precisely when G is 2-transitive. We then extend results of I. Pak on mixing times for a random walk on P(G). Our work depends on a new result for permutation groups involving…
Algorithmic decidability is established for two order-theoretic properties of downward closed subsets defined by finitely many obstructions in two infinite posets. The properties under consideration are: (a) being atomic, i.e. not being decomposable as a union of two downward closed proper subsets, or, equivalently, satisfying the joint embedding property; and (b) being well quasi-ordered. The two posets are: (1) words over a finite alphabet under the consecutive subword ordering; and (2) finite permutations under the consecutive subpermutation ordering. Underpinning the four results are chara…
The normalized volume of the Chan-Robbins-Yuen polytope ($CRY_n$) is the product of consecutive Catalan numbers. The polytope $CRY_n$ has captivated combinatorial audiences for over a decade, as there is no combinatorial proof for its volume formula. In their quest to understand $CRY_n$ better, the third author and Morales introduced two natural generalizations of it and conjectured that their volumes are certain powers of $2$ multiplied by a product of consecutive Catalan numbers. Zeilberger proved one of these conjectures. In this paper we present proofs of both conjectures.
We use various combinatorial and probabilistic techniques to study growth rates for the probability that a random permutation from the Mallows distribution avoids consecutive patterns. The Mallows distribution behaves like a $q$-analogue of the uniform distribution by weighting each permutation $π$ by $q^{inv(π)}$, where $inv(π)$ is the number of inversions in $π$ and $q$ is a positive, real-valued parameter. We prove that the growth rate exists for all patterns and all $q>0$, and we generalize Goulden and Jackson's cluster method to keep track of the number of inversions in permutations avoid…
The set of all permutations, ordered by pattern containment, is a poset. We give a formula for the Möbius function of intervals $[1,π]$ in this poset, for any permutation $π$ with at most one descent. We compute the Möbius function as a function of the number and positions of pairs of consecutive letters in $π$ that are consecutive in value. As a result of this we show that the Möbius function is unbounded on the poset of all permutations. We show that the Möbius function is zero on any interval $[1,π]$ where $π$ has a triple of consecutive letters whose values are consecutive and monotone. We…
We study polytopes defined by inequalities of the form $\sum_{i\in I} z_{i}\leq 1$ for $I\subseteq [d]$ and nonnegative $z_i$ where the inequalities can be reordered into a matrix inequality involving a column-convex $\{0,1\}$-matrix. These generalize polytopes studied by Stanley, and the consecutive coordinate polytopes of Ayyer, Josuat-Vergès, and Ramassamy. We prove an integral equivalence between these polytopes and flow polytopes of directed acyclic graphs $G$ with a Hamiltonian path, which we call spinal graphs. We show that the volume of these flow polytopes is the number of extensions …
Let $A^{(n)}_{l;k}\subset S_n$ denote the event that the set of $l$ consecutive numbers $\{k,k+1,\cdots, k+l-1\}$ appear in a set of $l$ consecutive positions. Let $p=\{p_j\}_{j=1}^\infty$ be a distribution on $\mathbb{N}$ with $p_j>0$. Let $P_n$ denote the probability measure on $S_n$ corresponding to the $p$-shifted random permutation. Our main result, under the additional assumption that $\{p_j\}_{j=1}^\infty$ is non-increasing, is that $$ \begin{aligned} &\lim_{l\to\infty}\lim_{n\to\infty}P_n(A^{(n )}_{l,k})=\big(\prod_{j=1}^{k-1}\sum_{i=1}^jp_i\big) \big(\prod_{j=1}^\infty\sum_{i=1}^jp_i\…
We set up a new notion of local convergence for permutations and we prove a characterization in terms of proportions of \emph{consecutive} pattern occurrences. We also characterize random limiting objects for this new topology introducing a notion of "shift-invariant" property (corresponding to the notion of unimodularity for random graphs). We then study two models in the framework of random pattern-avoiding permutations. We compute the local limits of uniform $ρ$-avoiding permutations, for $|ρ|=3,$ when the size of the permutations tends to infinity. The core part of the argument is the desc…
A positroid is a special case of a realizable matroid that arose from the study of the totally nonnegative part of the Grassmannian by Postnikov. In this paper, we study the facets of its matroid polytope and the independent set polytope. This allows one to describe the bases and independent sets directly from the decorated permutation, bypassing the use of the Grassmann necklace. We also describe a criterion for determining whether a given cyclic interval is a flat or not using the decorated permutation, then show how it applies to checking the concordancy of positroids.
Goulden and Jackson introduced a very powerful method to study the distributions of certain consecutive patterns in permutations, words, and other combinatorial objects which is now called the cluster method. There are a number of natural classes of combinatorial objects which start with either permutations or words and add additional restrictions. These include up-down permutations, generalized Euler permutations, words with no consecutive repeated letters, Young tableaux, and non-backtracking random walks. We develop an extension of the cluster method which we call the {\em generalized clust…
An assosiahedron $\mathcal{K}^n$, known also as Stasheff polytope, is a multifaceted combinatorial object, which, in particular, can be realized as a convex hull of certain points in $\mathbf{R}^{n}$, forming $(n-1)$-dimensional polytope. A permutahedron $\mathcal{P}^n$ is a polytope of dimension $(n-1)$ in $\mathbf{R}^{n}$ with vertices forming various permutations of $n$-element set. There exist well-known orderings of vertices of $\mathcal{P}^n$ and $\mathcal{K}^n$ that make these objects into lattices: the first known as permutation lattices, and the latter as Tamari lattices. We establish…
We consider the problem of enumerating permutations in the symmetric group on $n$ elements which avoid a given set of consecutive pattern $S$, and in particular computing asymptotics as $n$ tends to infinity. We develop a general method which solves this enumeration problem using the spectral theory of integral operators on $L^{2}([0,1]^{m})$, where the patterns in $S$ has length $m+1$. Kre\uın and Rutman's generalization of the Perron--Frobenius theory of non-negative matrices plays a central role. Our methods give detailed asymptotic expansions and allow for explicit computation of leading t…
Let $n$ and $k$ be positive integers with $n>k$. Given a permutation $(π_1,\ldots,π_n)$ of integers $1,\ldots,n$, we consider $k$-consecutive sums of $π$, i.e., $s_i:=\sum_{j=0}^{k-1}π_{i+j}$ for $i=1,\ldots,n$, where we let $π_{n+j}=π_j$. What we want to do in this paper is to know the exact value of $$\mathrm{msum}(n,k):=\min\left\{\max\{s_i : i=1,\ldots,n\} -\frac{k(n+1)}{2}: π\in S_n\right\},$$ where $S_n$ denotes the set of all permutations of $1,\ldots,n$. In this paper, we determine the exact values of $\mathrm{msum}(n,k)$ for some particular cases of $n$ and $k$. As a corollary of the …
We use the cluster method to enumerate permutations avoiding consecutive patterns. We reprove and generalize in a unified way several known results and obtain new ones, including some patterns of length 4 and 5, as well as some infinite families of patterns of a given shape. By enumerating linear extensions of certain posets, we find a differential equation satisfied by the inverse of the exponential generating function counting occurrences of the pattern. We prove that for a large class of patterns, this inverse is always an entire function. We also complete the classification of consecutive …
Motivated by the recent proof of the Stanley-Wilf conjecture, we study the asymptotic behavior of the number of permutations avoiding a generalized pattern. Generalized patterns allow the requirement that some pairs of letters must be adjacent in an occurrence of the pattern in the permutation, and consecutive patterns are a particular case of them. We determine the asymptotic behavior of the number of permutations avoiding a consecutive pattern, showing that they are an exponentially small proportion of the total number of permutations. For some other generalized patterns we give partial resu…
We study the number of values taken by the sums $\sum_{i=u}^{v-1} a_i$, where $a_1,a_2,\dots,a_n$ is a permutation of $1,2,\dots,n$ and $1 \leq u < v \leq n+1$. In particular, we show that for a random choice of a permutation, with high probability there are $(\frac{1+e^{-2}}{4} +o(1)) n^2$ such sums. This answers an old question of Erdős and Harzheim. We also obtain non-trivial bounds on the maximum possible number of distinct sums, ranging over all permutations of $1,2,\dots,n$. We close with some questions concerning the minimal possible number of distinct sums.
There are several approaches to study occurrences of consecutive patterns in permutations such as the inclusion-exclusion method, the tree representations of permutations, the spectral approach and others. We propose yet another approach to study occurrences of consecutive patterns in permutations. The approach is based on considering the graph of patterns overlaps, which is a certain subgraph of the de Bruijn graph. While applying our approach, the notion of a uniquely $k$-determined permutation appears. We give two criteria for a permutation to be uniquely $k$-determined: one in terms of the…
Descents in permutations or words are defined from the relative position of two consecutive letters. We investigate a statistic involving patterns of k consecutive letters, and show that it leads to Hopf algebras generalizing noncommutative symmetric functions and quasi-symmetric functions.
An occurrence of a consecutive permutation pattern $p$ in a permutation $π$ is a segment of consecutive letters of $π$ whose values appear in the same order of size as the letters in $p$. The set of all permutations forms a poset with respect to such pattern containment. We compute the Möbius function of intervals in this poset, providing what may be called a complete solution to the problem. For most intervals our results give an immediate answer to the question. In the remaining cases, we give a polynomial time algorithm to compute the Möbius function. In particular, we show that the Möbius …
We give multiple proofs of two formulas concerning the enumeration of permutations avoiding a monotone consecutive pattern with a certain value for the inverse peak number or inverse left peak number statistic. The enumeration in both cases is given by a sequence related to Fibonacci numbers. We also show that there is exactly one permutation whose inverse peak number is zero among all permutations with any fixed descent composition, and we give a few elementary consequences of this fact. Our proofs involve generating functions, symmetric functions, regular expressions, and monomino-domino til…
In this paper, we study the distribution of consecutive patterns in the set of 123-avoiding permutations and the set of 132-avoiding permutations, that is, in $\mathcal{S}_n(123)$ and $\mathcal{S}_n(132)$. We first study the distribution of consecutive pattern $γ$-matches in $\mathcal{S}_n(123)$ and $\mathcal{S}_n(132)$ for each length 3 consecutive pattern $γ$. Then we extend our methods to study the joint distributions of multiple consecutive patterns. Some more general cases are discussed in this paper as well.
We introduce vincular pattern posets, then we consider in particular the quasi-consecutive pattern poset, which is defined by declaring $σ\leq τ$ whenever the permutation $τ$ contains an occurrence of the permutation $σ$ in which all the entries are adjacent in $τ$ except at most the first and the second. We investigate the Möbius function of the quasi-consecutive pattern poset and we completely determine it for those intervals $[σ,τ]$ such that $σ$ occurs precisely once in $τ$.
Two permutations $π$ and $τ$ are c-Wilf equivalent if, for each $n$, the number of permutations in $S_n$ avoiding $π$ as a consecutive pattern (i.e., in adjacent positions) is the same as the number of those avoiding $τ$. In addition, $π$ and $τ$ are strongly c-Wilf equivalent if, for each $n$ and $k$, the number of permutations in $S_n$ containing $k$ occurrences of $π$ as a consecutive pattern is the same as for $τ$. In this paper we introduce a third, more restrictive equivalence relation, defining $π$ and $τ$ to be super-strongly c-Wilf equivalent if the above condition holds for any set o…
We consider two polytopes. The quadratic assignment polytope $QAP(n)$ is the convex hull of the set of tensors $x\otimes x$, $x \in P_n$, where $P_n$ is the set of $n\times n$ permutation matrices. The second polytope is defined as follows. For every permutation of vertices of the complete graph $K_n$ we consider appropriate $\binom{n}{2} \times \binom{n}{2}$ permutation matrix of the edges of $K_n$. The Young polytope $P((n-2,2))$ is the convex hull of all such matrices. In 2009, S. Onn showed that the subgraph isomorphism problem can be reduced to optimization both over $QAP(n)$ and over $P(…
The Birkhoff polytope $B_n$ is the convex hull of all $n\times n$ permutation matrices in $\mathbb{R}^{n\times n}$. We compute the combinatorial symmetry group of the Birkhoff polytope. A representation polytope is the convex hull of some finite matrix group $G\leq \operatorname{GL}(d,\mathbb{R})$. We show that the group of permutation matrices is essentially the only finite matrix group which yields a representation polytope with the same face lattice as the Birkhoff polytope.
We prove that the number of permutations avoiding an arbitrary consecutive pattern of length m is asymptotically largest when the avoided pattern is 12...m, and smallest when the avoided pattern is 12...(m-2)m(m-1). This settles a conjecture of the author and Noy from 2001, as well as another recent conjecture of Nakamura. We also show that among non-overlapping patterns of length m, the pattern 134...m2 is the one for which the number of permutations avoiding it is asymptotically largest.
The Birkhoff polytope, defined to be the convex hull of $n\times n$ permutation matrices, is a well studied polytope in the context of the Ehrhart theory. This polytope is known to have many desirable properties, such as the Gorenstein property and existence of regular, unimodular triangulations. In this paper, we study analogues of the Birkhoff polytope for finite irreducible Coxeter groups of other types. We focus on a type-$B$ Birkhoff polytope $\mathcal{BB}(n)$ arising from signed permutation matrices and prove that it and its dual polytope are reflexive, and hence Gorenstein, and also pos…
