Austin Allen, Dylan Cruz Fonseca, Veronica Dobbs, Egypt Downs, Evelyn Fokuoh, Anant Godbole, Sebastián Papanikolaou Costa, Christopher Soto, Lino Yoshikawa
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.
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Arrangements of Consecutive Numbers in Mallows Permutations
研究Mallows分布下排列中连续数字聚类排列的计数随机变量，给出了期望的渐近表达式，并确定了分布近似泊松分布的参数范围。
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 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…
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 sequ…
An Alternative Proof for the Expected Number of Distinct Consecutive Patterns in a Random Permutation
本文用 Stein-Chen Poisson 逼近方法重新证明并略微改进了随机排列中不同连续模式期望数的渐近结果。
The feasible regions for consecutive patterns of pattern-avoiding permutations
本研究确定模式回避排列族连续模式可行区域的维数、凸性及多面体性质，并给出特定情形下顶点描述，连接模式堆积与局部极限问题。
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 $n \geq 2$.
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…
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.
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 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.
A Lower Bound on the Expected Number of Distinct Patterns in a Random Permutation
本文研究随机排列中不同非连续模式的期望数量，证明其至少为2^{n-1}(1+o(1))，推翻了非连续模式的猜想。
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…
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 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 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…
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…
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…
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.
Consecutive Pattern Containment and c-Wilf Equivalence
本文用初等方法重新证明连续模式包含的已有结果，并给出排列中连续模式避免增长率的新一般界。
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…
Maximal Lehmer Codes for Permutations with Classical and Consecutive 321-Avoidance
本文利用Lehmer码构造加权偏序集，研究避免经典和连续321模式的排列，确定了极大元素数量及最大权重元素的存在性，对应最大逆序数的排列。
Emerging consecutive pattern avoidance
本文研究了避免至少两个长度为3连续模式的排列类中的渐近流行度，通过解析与双射方法详细推导了部分类别的极限概率，并指出五类问题仍未解决。
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.
Periodic Points of Consecutive-Pattern-Avoiding Stack-Sorting Maps
本文研究了连续模式规避栈排序映射 SC_σ 的动力学性质，证明了其周期点恰为连续规避 σ 及其逆序排列的排列，从而解决了 Defant 和 Zheng 提出的主要猜想。
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…
Clustering of consecutive numbers in permutations avoiding a pattern of length three or avoiding a finite number of simple patterns
本文研究避免三类模式或有限个简单模式的排列中连续数字相邻聚集的期望次数，给出精确计数公式及两类渐近结果，并涵盖可分排列。
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.
Enumeration and Distribution of Permutation Rows and Columns in Equi-$n$-Squares
本文研究了等 $n$-方阵中排列行和列的计数与分布，推导了其数量的精确和渐近公式，并通过模拟验证了理论结果。
We extend the reciprocity method of Jones and Remmel to study generating functions of the form $$\sum_{n \geq 0} \frac{t^n}{n!} \sum_{σ\in \mathcal{NM}_n(Γ)}x^{\mathrm{LRmin}(σ)}y^{1+\mathrm{des}(σ)}$$ where $Γ$ is a set of permutations which start with 1 and have at most one descent, $\mathcal{NM}_n(Γ)$ is the set of permutations $σ$ in the symmetric group $\mathfrak{S}_n$ which have no $Γ$-matches, $\mathrm{des}(σ)$ is the number of descents of $σ$ and $\mathrm{LRmin}(σ)$ is the number of left-to-right minima of $σ$. We show that this generating function is of the form $\left( \frac{1}{U_Γ(t…
Let $S_n$ denote the group all permutations of $n$. For every permutation $σ$, we let $\mathrm{des}(σ)$ denote the number of descents in $σ$ and $\mathrm{LRMin}(σ)$ denote the number of left-to-right minima of $σ$. Given a sequence $τ= τ_1 \cdots τ_n$ of distinct positive integers, we define the reduction of $τ$, $\mathrm{red}(τ)$, to be the permutation of $S_n$ that results by replacing the $i$-th smallest element of $τ$ by $i$. If $Γ$ is a set of permutations, we say that a permutation $σ= σ_1 \ldots σ_n \in S_n$ has a $Γ$-match starting at position $i$ if there is a $i < j$ such that $\math…
Real-rooted Eulerian polynomials from permutations, words, and paths
本文研究六个Eulerian型多项式族，证明错排下降多项式实根性等猜想，并加强或解决多个相关实根性与交错性问题。
A grid polygon is a polygon whose vertices are points of a grid. We define an injective map between permutations of length n and a subset of grid polygons on n vertices, which we call consecutive-minima polygons. By the kernel method, we enumerate sets of permutations whose consecutive-minima polygons satisfy specific geometric conditions. We deal with 2-variate and 3-variate generating functions involving derivatives, cases which are not routinely solved by the kernel method.
Computational Approach to the $SC_{231}$ Consecutive-Pattern-Avoiding Stack Sort
本文计算了长度不超过14的排列的排序数，并估计了长度1000内平均排序数，发现最大和平均排序数随n增长快于线性，同时证明了$SC_{231}$的数学性质，如最大排序数下界和上界。
Consecutive patterns in restricted permutations and involutions
本文通过 Foata 映射和 Laguerre 历史双射，研究避免特定连续与经典模式的排列及受限对合，确定 des 与 inv 的联合分布及所有长度三连续模式的分布。
Elizalde (2011) characterized which permutations can be obtained by ordering consecutive elements in the trajectories of (positive) beta-transformations and beta-shifts. We prove similar results for negative bases beta.
We introduce consecutive-pattern-avoiding stack-sorting maps $\text{SC}_σ$, which are natural generalizations of West's stack-sorting map $s$ and natural analogues of the classical-pattern-avoiding stack-sorting maps $s_σ$ recently introduced by Cerbai, Claesson, and Ferrari. We characterize the patterns $σ$ such that $\text{Sort}(\text{SC}_σ)$, the set of permutations that are sortable via the map $s\circ\text{SC}_σ$, is a permutation class, and we enumerate the sets $\text{Sort}(\text{SC}_σ)$ for $σ\in\{123,132,321\}$. We also study the maps $\text{SC}_σ$ from a dynamical point of view, char…
We introduce a new permutation statistic, namely, the number of cycles of length $q$ consisting of consecutive integers, and consider the distribution of this statistic among the permutations of $\{1,2,...,n\}$. We determine explicit formulas, recurrence relations, and ordinary and exponential generating functions. A generalization to more than one fixed length is also considered.
We describe the limit (for two topologies) of large uniform random square permutations, i.e., permutations where every point is a record. The starting point for all our results is a sampling procedure for asymptotically uniform square permutations. Building on that, we first describe the global behavior by showing that these permutations have a permuton limit which can be described by a random rectangle. We also explore fluctuations about this random rectangle, which we can describe through coupled Brownian motions. Second, we consider the limiting behavior of the neighborhood of a point in th…
On consecutive pattern-avoiding permutations of length 4, 5 and beyond
本文研究长度为4、5及更长的连续模式避免排列的生成函数，给出新解、递推和代数解，并提出更快的多项式时间算法，以更高精度计算渐近性，推测未解决类非D-有限。
We provide a bijective proof of a formula of Auli and the author expressing the number of inversion sequences with no three consecutive equal entries in terms of the number of non-derangements, that is, permutations with fixed points. Additionally, we give bijective proofs of two simple recurrences for the number of non-derangements.
Pattern Avoidance for Random Permutations
本文利用泊松逼近方法，为随机排列中任意模式回避计数提供显式误差界，并推广至马洛斯排列的连续模式，估计模式出现概率。
We show that if $A=\{a_1 < a_2 < \ldots < a_k\}$ is a set of real numbers such that the differences of the consecutive elements are distinct, then for and finite $B \subset \mathbb{R}$, $$|A+B|\gg |A|^{1/2}|B|.$$ The bound is tight up to the constant.
For permutations avoiding consecutive patterns from a given set, we present a combinatorial formula for the multiplicative inverse of the corresponding exponential generating function. The formula comes from homological algebra considerations in the same sense as the corresponding inversion formula for avoiding word patterns comes from the well known Anick's resolution.
In this article we define an encoding for parabolic permutations that distinguishes between parabolic $231$-avoiding permutations. We prove that the componentwise order on these codes realizes the parabolic Tamari lattice, and conclude a direct and simple proof that the parabolic Tamari lattice is isomorphic to a certain $ν$-Tamari lattice, with an explicit bijection. Furthermore, we prove that this bijection is closely related to the map $Θ$ used when the lattice isomorphism was first proved in (Ceballos, Fang and Mühle, 2020), settling an open problem therein.
Shuffle algebras are monoids for an unconvential monoidal category structure on graded vector spaces. We present two homological results on shuffle algebras with monomial relations, and use them to prove exact and asymptotic results on consecutive pattern avoidance in permutations.
Using the approach suggested in [arXiv:1002.2761] we present below a sufficient condition guaranteeing that two collections of patterns of permutations have the same exponential generating functions for the number of permutations avoiding elements of these collections as consecutive patterns. In short, the coincidence of the latter generating functions is guaranteed by a length-preserving bijection of patterns in these collections which is identical on the overlappings of pairs of patterns where the overlappings are considered as unordered sets. Our proof is based on a direct algorithm for the…
A permutation is so-called two stack sortable if it (i) avoids the (scattered) pattern 2-3-4-1, and (ii) contains a 3-2-4-1 pattern only as part of a 3-5-2-4-1 pattern. Here we show that the permutations on [n] satisfying condition (ii) alone are equinumerous with the permutations on [n] that avoid the mixed scattered/consecutive pattern 31-4-2. The proof uses a known bijection from 3-2-1-avoiding to 3-1-2-avoiding permutations.
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