Permutations avoiding 4321 and 3241 have an algebraic generating function
David Callan
本文证明避免模式4321和3241的排列的生成函数是代数的，证实了Kruchinin的猜想。
我们证明了同时避免（经典）模式4321和3241的排列具有由Vladimir Kruchinin所 conjectured 的代数生成函数。
We show that permutations avoiding both of the (classical) patterns 4321 and 3241 have the algebraic generating function conjectured by Vladimir Kruchinin.
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We investigate the notion of almost avoiding a permutation: $π$ almost avoids $β$ if one can remove a single entry from $π$ to obtain a $β$-avoiding permutation.
Refining an existing counting argument, we provide an improved upper bound for the number of 1324-avoiding permutations of a given length.
We define a bijection between triangulations of a convex polygon and $312$-avoiding permutations through the process of "ear-clipping". This bijection is then used to obtain a bijection between polygon dissections and a certain class of $312$-avoiding permutations.
On the diagram of 132-avoiding permutations
避免132的置换的图可以很容易地被描述：它只是一个划分的图。基于这一事实，我们提出了一个新的双射之间的132-避免和321-避免排列。我们将表明，这种双射翻译这些排列和戴克路径之间的对应关系，分别由Krattenthaler和Billey-Jockusch-Stanley，彼此。此外，图的方法得到简单的证明，一些枚举结果有关的禁止模式在132-避免排列。
We provide a simple injective proof that the number of 132-avoiding permutations with a unique longest increasing subsequence is at least as large as the number of 132-avoiding permutations without a unique longest increasing subsequence.
Alternating, pattern-avoiding permutations
研究了避免包含132个置换模式的交替置换的计数问题。我们在132-避免排列的集合S_n（132）和交替的132-避免排列的集合A_{2n + 1}（132）之间构造了一个双射。对于每个集合p_1，...，p_k的模式和某些相关模式q_1，.，q_k，我们的双射限制为S_n（132，p_1，...，p_k）、避免132和p_i的置换集合，以及A_{2n + 1}（132，q_1，.，q_k）、交替置换避免集合132和q_i.这将后一个集合的枚举减少为前一个集合的枚举。
A Complete Enumeration of Ballot Permutations Avoiding Sets of Small Patterns
本文完全枚举了避免两个或三个长度为3模式的选票排列，建立了相应的递推关系与公式，并发现了避免132和312的选票排列与Dyck路径左因子间的双射，同时完成了相关的Wilf分类。
We prove that the number of 1324-avoiding permutations of length n is less than (7+4\sqrt{3})^n.
We modify the enumeration schemes of Zeilberger and Vatter so that they can efficiently enumerate many new classes of pattern-avoiding permutations including all such classes with a regular insertion encoding.
We enumerate the pattern class Av(2143,4231) and completely describe its permutations. The main tools are simple permutations and monotone grid classes.
We answer a question of R. J. Mathar and confirm that the counting sequence for $\bar{2}413\bar{5}$-avoiding permutations is the Invert transform of the Bell numbers. The proof relies on a simple decomposition of these permutations and the known fact that $\bar{2}413$-avoiding permutations are counted by the Bell numbers.
We show a $n^2 \cdot 2^{n/2}$ upper bound on the number of $(132,213)$ avoiding cyclic permutations. This is the first nontrivial upper bound on the number of such permutations. We also construct an algorithm to determine whether a $(132,213)$ avoiding permutation is cyclic that references only the permutation's layer lengths.
We enumerate permutations that avoid all but one of the $k$ patterns of length $k$ starting with a monotone increasing subsequence of length $k-1$. We compare the size of such permutation classes to the size of the class of permutations avoiding the monotone increasing subsequence of length $k-1$.
Pattern avoiding permutations are context-sensitive
我们建立一个双射从所有的排列（给定长度），避免了模式$q$和上下文敏感的语言。
A permutation is (1-23-4)-avoiding if it contains no four entries, increasing left to right, with the middle two adjacent in the permutation. Here we give a 2-variable recurrence for the number of such permutations, improving on the previously known 4-variable recurrence. At the heart of the proof is a bijection from (1-23-4)-avoiding permutations to increasing ordered trees whose leaves, taken in preorder, are also increasing.
On the Wilf-Stanley limit of 4231-avoiding permutations and a conjecture of Arratia
我们构造了一个有限自动机序列，它接受4231-避免置换类的子类。因此，我们证明了4231-避免置换类的Wilf-Stanley极限在9.35以下有界。这个界表明，在所有避免长度为4的单个置换的置换类中，这类置换具有最大的这种极限，并且驳斥了关于避免长度为k的单个置换的置换类的威尔夫-斯坦利极限不能超过（k-1）^2的猜想。
We construct a bijection between 231-avoiding permutations and Dyck paths that sends the sum of the major index and the inverse major index of a 231-avoiding permutation to the major index of the corresponding Dyck path. Furthermore, we relate this bijection to others and exhibit a bistatistic on 231-avoiding permutations which is related to the q,t-Catalan numbers.
Restricted 132-avoiding permutations
我们研究生成函数的排列数n个字母避免132和任意排列$τ$的k个字母，或包含$τ$正好一次。在一些有趣的情况下，生成函数只依赖于k，并通过第二类切比雪夫多项式表示。
We present some combinatorial interpretations for coefficients appearing in series partitioning the permutations avoiding 132 along marked mesh patterns. We identify for patterns in which only one parameter is non zero the combinatorial family in bijection with 132-avoiding permutations and also preserving the statistic counted by the marked mesh pattern.
We show that permutations of size $n$ avoiding both of the dashed patterns 32-41 and 41-32 are equinumerous with indecomposable set partitions of size $n+1$, and deduce a related result.
We consider the distribution of ascents, descents, peaks, valleys, double ascents, and double descents over permutations avoiding a set of patterns. Many of these statistics have already been studied over sets of permutations avoiding a single pattern of length 3. However, the distribution of peaks over 321-avoiding permutations is new and we relate it statistics on Dyck paths. We also obtain new interpretations of a number of well-known combinatorial sequences by studying these statistics over permutations avoiding two patterns of length 3.
We investigate permutations and involutions that avoid a pattern of length three and have a {\em unique} longest increasing subsequence.
Shape Avoiding Permutations
排列避免所有模式的一个给定的形状（在意义上的Robinson-Schensted-Knuth）。我们证明了所有这样的排列的形状都包含在一个适当的厚钩中，并推导出它们的数量的指数增长性。
Enumeration of Dumont permutations avoiding certain four-letter patterns
本文枚举了第四类Dumont排列中避免或包含特定四字母模式的情况，并猜想第一类Dumont排列上两个四字母模式的Wilf等价性。
We exploit Krattenthaler's bijection between 123-avoiding permutations and Dyck paths to determine the Eulerian distribution over the set $S_n(123)$ of 123-avoiding permutations in $S_n$. In particular, we show that the descents of a permutation correspond to valleys and triple falls of the associated Dyck path. We get the Eulerian numbers of $S_n(123)$ by studying the joint distribution of these two statistics on Dyck paths.
Groups generated by pattern avoiding permutations
本文研究由避免给定模式的排列集合生成的群，先给出其结构的一般性定理，再对避免少量短模式的情形逐类分析。
Some combinatorics related to central binomial coefficients: Grand-Dyck paths, coloured noncrossing partitions and signed pattern avoiding permutations
我们给出了某些整数序列的参数的Grand-Dyck路径和有色不相交划分的一些解释，我们发现了一些新的双射有关Grand-Dyck路径和符号模式避免置换。接下来，我们转移一个自然的分配格结构的Grand-Dyck路径的有色非交叉分区和签署的模式，避免置换，从而显示，特别是，它是同构的结构诱导的（强）Bruhat秩序上的某一组签署的模式，避免置换。
We find the generating function for the class of all permutations that avoid the patterns 3124 and 4312 by showing that it is an inflation of the union of two geometric grid classes.
Pattern avoiding machines were recently introduced by Claesson, Ferrari and the current author to gain a better understanding of the classical $2$-stacksort problem. In this paper we generalize these devices by allowing permutations with repeated elements, also known as Cayley permutations. The main result is a description of those patterns such that the corresponding set of sortable permutations is a class. We also show a new involution on the set of Cayley permutations, obtained by regarding a pattern-avoiding stack as an operator. Finally, we analyze two generalizations of pop-stack sorting…
Babson and Steingrìmsson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Subsequently, Claesson presented a complete solution for the number of permutations avoiding any single pattern of type (1,2) or (2,1). For eight of these twelve patterns the answer is given by the Bell numbers. For the remaining four the answer is given by the Catalan numbers. In the present paper we give a complete solution for the number of permutations avoiding a pair of patterns of type (1,2) or (2,1). We also conjectur…
Asymptotic enumeration of permutations avoiding generalized patterns
受最近Stanley-Wilf猜想证明的启发，我们研究了避免广义模式的排列数的渐近行为。广义模式允许要求某些字母对在排列中的模式出现时必须相邻，并且连续模式是它们的特殊情况。我们确定的渐近行为的排列避免连续模式的数量，表明它们是一个指数小的比例排列的总数。对于其他一些广义模式，我们给出了部分结果，表明避免它们的排列数的增长速度比经典模式，但比连续模式更慢。
This paper is continuation of the study of the 1-box pattern in permutations introduced by the authors in \cite{kitrem4}. We derive a two-variable generating function for the distribution of this pattern on 132-avoiding permutations, and then study some of its coefficients providing a link to the Fibonacci numbers. We also find the number of separable permutations with two and three occurrences of the 1-box pattern.
The simple permutations in two permutation classes --- the 321-avoiding permutations and the skew-merged permutations --- are enumerated using a uniform method. In both cases, these enumerations were known implicitly, by working backwards from the enumeration of the class, but the simple permutations had not been enumerated explicitly. In particular, the enumeration of the simple skew-merged permutations leads to the first truly structural enumeration of this class as a whole. The extension of this method to a wider collection of classes namely grid classes of infinite paths is discussed.
Comtet introduced the notion of indecomposable permutations in 1972. A permutation is indecomposable if and only if it has no proper prefix which is itself a permutation. Indecomposable permutations were studied in the literature in various contexts. In particular, this notion has been proven to be useful in obtaining non-trivial enumeration and equidistribution results on permutations. In this paper, we give a complete classification of indecomposable permutations avoiding a classical pattern of length 3 or 4, and of indecomposable permutations avoiding a non-consecutive vincular pattern of l…
In a previous paper, we showed that $3\bar{5}241$-avoiding permutations are counted by the unique sequence that starts with a 1 and shifts left under the self-composition transform. The proof uses a complicated bijection. Here we give a much simpler proof based on Lagrange inversion.
We define a map $ν$ between the symmetric group $S_n$ and the set of pairs of Dyck paths of semilength $n$. We show that the map $ν$ is injective when restricted to the set of 1234-avoiding permutations and characterize the image of this map.
Some statistics on permutations avoiding generalized patterns
在过去的十年中，大量的文章已经发表，研究模式避免排列。从枚举的角度来看，通常人们试图根据其长度来计算避免某些模式的排列。在这里，我们通过考虑统计数据的“第一个/最后一个条目”来解决细化这个枚举的问题。我们给出了$(1,2)$或$(2,1)$类型的广义模式的完整结果，以及避免上述类型的广义模式对的置换的某些情况。
We introduce a new method for encoding permutations as weighted independent sets in a family of graphs we call cores. The encoding allows us to enumerate (1324, 2143)-, (1234, 1324, 2143)-, (1234, 1324, 1432, 3214)-avoiding permutations with respect to their number of "boundary points" and the size of the independent set in the graph they correspond to.
We give a natural definition of rowmotion for $321$-avoiding permutations, by translating, through bijections involving Dyck paths and the Lalanne--Kreweras involution, the analogous notion for antichains of the positive root poset of type $A$. We prove that some permutation statistics, such as the number of fixed points, are homomesic under rowmotion, meaning that they have a constant average over its orbits. Our setting also provides a more natural description of the celebrated Armstrong--Stump--Thomas equivariant bijection between antichains and non-crossing matchings in types $A$ and $B$, …
This addendum contains results about the inversion number and major index polynomials for permutations avoiding 321 which did not fit well into the original paper. In particular, we consider symmetry, unimodality, behavior modulo 2, and signed enumeration.
Permutations avoiding a set of patterns from S_3 and a pattern from S_4
本文计算了S_n中避开S_4的一个模式和避开S_3的一个非空模式集的所有置换的集合S_n（T，τ）的基数。
We prove a generalization of a conjecture of Dokos, Dwyer, Johnson, Sagan, and Selsor giving a recursion for the inversion polynomial of 321-avoiding permutations. We also answer a question they posed about finding a recursive formulas for the major index polynomial of 321-avoiding permutations. Other properties of these polynomials are investigated as well. Our tools include Dyck and 2-Motzkin paths, polyominoes, and continued fractions.
A centrosymmetric permutation is one which is invariant under the reverse-complement operation, or equivalently one whose associated standard Young tableaux under the Robinson-Schensted algorithm are both invariant under the Schutzenberger involution. In this paper, we characterize the set of permutations avoiding 1243 and 2143 whose images under the reverse-complement mapping also avoid these patterns. We also characterize in a simple manner the corresponding Schroder paths under a bijection of Egge and Mansour. We then use these results to enumerate centrosymmetric permutations avoiding the …
Pattern-avoiding shallow permutations
本文研究了所有$σ\in \mathcal{S}_3$的避开$σ$的浅排列，通过开发更一般的结构结果，成功枚举了避开任何长度为3模式的浅排列。
Pattern-avoiding stabilized-interval-free permutations
本文研究无稳定区间（SIF）排列对长度3经典模式及模式对的避免问题，利用不动点分布、组合论证及与有序森林的双射完成枚举，并提出相关猜想。
Enumerating permutations avoiding more than three Babson - Steingr\'ımsson patterns
不久前，Claesson和Mansour提出了避免多于三个Babson-Steingr msson模式（$(1,2)$或$(2,1)$类型的广义模式）的排列计数的一些定理.已经考虑了避免一种、两种或三种模式。在这里，四个和五个禁止模式的情况下，解决了和避免他们的排列的精确计数，确认Claesson和Mansour的命题。我们使用的方法可以很容易地扩展到五个以上的禁止模式的情况下。
(32-1)-Avoiding Permutations with Maximum Inversion Number
研究(32 - 1) - 避免排列，确定其最大逆序数并计数，给出直接构造来枚举达到最大逆序数的排列。
Restricted Dumont permutations, Dyck paths, and noncrossing partitions
我们完成了第二类杜蒙排列的枚举，避免了长度为4的模式，它本身就是第二类杜蒙排列。我们还考虑了一些组合统计Dumont排列避免某些模式的长度为3和4，并给出了一个自然的双射之间的3142-避免Dumont排列的第二类和非交叉分区，使用循环分解，以及双射之间的132-，231-和321-避免Dumont排列和Dyck路径。最后，我们列举杜蒙排列的第一类同时避免某些对4字母的模式和另一种模式的任意长度。
A permutation $σ\in\mathfrak{S}_n$ is simsun if for all $k$, the subword of $σ$ restricted to $\{1,...,k\}$ does not have three consecutive decreasing elements. The permutation $σ$ is double simsun if both $σ$ and $σ^{-1}$ are simsun. In this paper we present a new bijection between simsun permutations and increasing 1-2 trees, and show a number of interesting consequences of this bijection in the enumeration of pattern-avoiding simsun and double simsun permutations. We also enumerate the double simsun permutations that avoid each pattern of length three.
Numerical evidence suggests that certain permutation patterns of length k are easier to avoid than any other patterns of that same length. We prove that these patterns are avoided by no more than (2.25k^2)^n permutations of length n. In light of this, we conjecture that no pattern of length k is avoided by more than that many permutations of length n.
Permutations Containing and Avoiding 123 and 132 Patterns
证明了避免132-模式而恰好有一个123-模式的排列数等于（n-2）2^（n-3）。然后，我们给出了双射到一组排列，避免123-模式，只有一个132-模式。最后证明了包含一个123-模式和一个132-模式的排列的个数为（n-3）（n-4）2^（n-5）。
On the length of the longest subsequence avoiding an arbitrary pattern in a random permutation
我们考虑了在长度为n的随机置换中，避开给定模式的最长子序列的长度分布。最长递增子序列的充分研究的情况对应于避免模式21。我们表明，有一些常数c，使这个长度的平均值是渐近的两倍的平方根的c倍n和长度的分布是紧密集中在其平均值。我们观察到一些明显的连接c和斯坦利-威尔夫限制类的排列，避免给定的模式。
We introduce a new concept of permutation avoidance pattern called hatted pattern, which is a natural generalization of the barred pattern. We show the growth rate of the class of permutations avoiding a hatted pattern in comparison to barred pattern. We prove that Dyck paths with no peak at height $p$, Dyck paths with no $ud... du$ and Motzkin paths are counted by hatted pattern avoiding permutations in $\s_n(132)$ by showing explicit bijections. As a result, a new direct bijection between Motzkin paths and permutations in $\s_n(132)$ without two consecutive adjacent numbers is given. These p…
Permutations avoiding bipartite partially ordered patterns have a regular insertion encoding
该研究证明避免高度≤2的偏序模式（POP）的排列类有正则插入编码及有理生成函数，还通过组合探索得到数百个避免大小为5的POP的排列类的组合规范与生成函数，解决了多个相关猜想。
Refined Restricted Permutations Avoiding Subsets of Patterns of Length Three
定义$S_n^k(T)$为$\{1,2,...,n\}$的置换集，其中$k$不动点恰好避免了$T \subseteq S_m$中的所有模式。我们列举了所有$|T| \geq 2$和$0 \leq k \leq n$的$S_n^k(T)$、$T \subseteq S_3$。
A bijection between $321$- and $213$-avoiding permutations preserving $t$-stack-sortability
本文构造了保持$t$-stack-sortability的$321$-和$213$-避免排列之间的双射，以递增二叉树为核心方法证明了Zhang和Kitaev计数猜想的细化。
Lehmer Codes and the Reverse-Complement Mapping from (32-1)-Avoiding Permutations to (3-21)-Avoiding Permutations
该研究以避免特定vincular模式的排列为对象，借助莱默码将其转化为加权偏序集，证明其极大元由斐波那契数计数，且经典逆补映射在极大元间构成双射，揭示了偏序集结构的对称性。
132-avoiding Two-stack Sortable Permutations, Fibonacci Numbers, and Pell Numbers
在1990年，West证明了在$n$字母上存在$2(3n)!/((n+1)!(2n+1)!)$两栈可排序排列。这个猜想是由Zeilberger在1992年证明的。后来，Dulucq、Gire和Guibert给出了这个猜想的组合证明。在本文中，我们研究的生成函数的两个堆栈可排序的排列在$n$字母避免（或包含正好一次）132和避免（或包含正好一次）的任意排列$τ$在$k$字母。在几个有趣的情况下，这个生成函数可以用斐波那契数的生成函数或佩尔数的生成函数来表示。
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.
Counting permutations avoiding two flat partially ordered patterns
研究同时避免两个平坦部分有序模式的排列计数问题，通过建立与k-斐波那契数的联系和双射方法，推导生成函数并扩展了单个平坦POPs的计数结果。
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