Pattern avoiding permutations are context-sensitive

避免小模式集合的选票排列的完全枚举

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。

A new upper bound for 1324-avoiding permutations

We prove that the number of 1324-avoiding permutations of length n is less than (7+4\sqrt{3})^n.

Flexible Schemes for Pattern-Avoiding Permutations

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.

The enumeration of permutations avoiding 2143 and 4231

We enumerate the pattern class Av(2143,4231) and completely describe its permutations. The main tools are simple permutations and monotone grid classes.

The number of \bar{2}413\bar{5}-avoiding permutations

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.

An Upper Bound on the Number of $(132,213)$-Avoiding Cyclic Permutations

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.

Permutations avoiding sets of patterns with long monotone subsequences

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 $k-1$.

A bijection to count (1-23-4)-avoiding permutations

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

We construct a sequence of finite automata that accept subclasses of the class of 4231-avoiding permutations. We thereby show that the Wilf-Stanley limit for the class of 4231-avoiding permutations is bounded below by 9.35. This bound shows that this class has the largest such limit among all classes of permutations avoiding a single permutation of length 4 and refutes the conjecture that the Wilf-Stanley limit of a class of permutations avoiding a single permutation of length k cannot exceed (k-1)^2.

On bijections between 231-avoiding permutations and Dyck paths

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

We study generating functions for the number of permutations on n letters avoiding 132 and an arbitrary permutation $τ$ on k letters, or containing $τ$ exactly once. In several interesting cases the generating function depends only on k and is expressed via Chebyshev polynomials of the second kind.

Combinatorics of simple marked mesh patterns in 132-avoiding permutations

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.

On permutations avoiding the dashed patterns 32-41 and 41-32

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.

Distributions of Statistics over Pattern-Avoiding Permutations

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.

Pattern avoiding permutations and involutions with a unique longest increasing subsequence

We investigate permutations and involutions that avoid a pattern of length three and have a {\em unique} longest increasing subsequence.

Shape Avoiding Permutations

Permutations avoiding all patterns of a given shape (in the sense of Robinson-Schensted-Knuth) are considered. We show that the shapes of all such permutations are contained in a suitable thick hook, and deduce an exponential growth rate for their number.

避免某些四字母模式的Dumont排列的枚举

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。

The descent statistic over 123-avoiding permutations

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.

由避免模式的排列生成的群

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

We give some interpretations to certain integer sequences in terms of parameters on Grand-Dyck paths and coloured noncrossing partitions, and we find some new bijections relating Grand-Dyck paths and signed pattern avoiding permutations. Next we transfer a natural distributive lattice structure on Grand-Dyck paths to coloured noncrossing partitions and signed pattern avoiding permutations, thus showing, in particular, that it is isomorphic to the structure induced by the (strong) Bruhat order on a certain set of signed pattern avoiding permutations.

The Enumeration of Permutations Avoiding 3124 and 4312

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.

Sorting Cayley permutations with pattern-avoiding machines

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…

Enumerating permutations avoiding a pair of Babson-Steingrimsson patterns

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

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…

The 1-box pattern on pattern avoiding permutations

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.

Generating and enumerating 321-avoiding and skew-merged simple permutations

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.

On pattern avoiding indecomposable permutations

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…

Lagrange Inversion Counts $3\bar{5}241$-Avoiding Permutations

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.

1234-avoiding permutations and Dyck paths

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

In the last decade a huge amount of articles has been published studying pattern avoidance on permutations. From the point of view of enumeration, typically one tries to count permutations avoiding certain patterns according to their lengths. Here we tackle the problem of refining this enumeration by considering the statistics "first/last entry". We give complete results for every generalized patterns of type $(1,2)$ or $(2,1)$ as well as for some cases of permutations avoiding a pair of generalized patterns of the above types.

Pattern avoiding permutations and independent sets in graphs

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.

Rowmotion on 321-avoiding permutations

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 $B$, …

Inversion polynomials for 321-avoiding permutations: addendum

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

In this paper we calculate the cardinality of the set S_n(T,tau) of all permutations in S_n that avoid one pattern from S_4 and a nonempty set of patterns from S_3.

Inversion polynomials for 321-avoiding permutations

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.

Centrosymmetric Permutations and Involutions Avoiding 1243 and 2143

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 …

避开模式的浅排列

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。

避免模式的无稳定区间排列

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。

避免4321和3241的排列具有代数生成函数

Permutations avoiding 4321 and 3241 have an algebraic generating function

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。

Enumerating permutations avoiding more than three Babson - Steingr\'ımsson patterns

Not long ago, Claesson and Mansour proposed some conjectures about the enumeration of the permutations avoiding more than three Babson - Steingr\'ımsson patterns (generalized patterns of type $(1,2)$ or $(2,1)$). The avoidance of one, two or three patterns has already been considered. Here, the cases of four and five forbidden patterns are solved and the exact enumeration of the permutations avoiding them is given, confirming the conjectures of Claesson and Mansour. The approach we use can be easily extended to the cases of more than five forbidden patterns.

(32 - 1) - 避免具有最大逆序数的排列

(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

We complete the enumeration of Dumont permutations of the second kind avoiding a pattern of length 4 which is itself a Dumont permutation of the second kind. We also consider some combinatorial statistics on Dumont permutations avoiding certain patterns of length 3 and 4 and give a natural bijection between 3142-avoiding Dumont permutations of the second kind and noncrossing partitions that uses cycle decomposition, as well as bijections between 132-, 231- and 321-avoiding Dumont permutations and Dyck paths. Finally, we enumerate Dumont permutations of the first kind simultaneously avoiding ce…

On Simsun and Double Simsun Permutations Avoiding a Pattern of Length Three

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.

On the Best Upper Bound for Permutations Avoiding A Pattern of a Given Length

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

We prove that the number of permutations which avoid 132-patterns and have exactly one 123-pattern equals (n-2)2^(n-3). We then give a bijection onto the set of permutations which avoid 123-patterns and have exactly one 132-pattern. Finally, we show that the number of permutations which contain exactly one 123-pattern and exactly one 132-pattern is (n-3)(n-4)2^(n-5).

On the length of the longest subsequence avoiding an arbitrary pattern in a random permutation

We consider the distribution of the length of the longest subsequence avoiding a given pattern in a random permutation of length n. The well-studied case of a longest increasing subsequence corresponds to avoiding the pattern 21. We show that there is some constant c such that the mean value of this length is asymptotic to twice the square root of c times n and that the distribution of the length is tightly concentrated around its mean. We observe some apparent connections between c and the Stanley-Wilf limit of the class of permutations avoiding the given pattern.

Combinatorial bijections from hatted avoiding permutations in $S_n(132)$ to generalized Dyck and Motzkin paths

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…

避免二部偏序模式的排列具有正则插入编码

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

Define $S_n^k(T)$ to be the set of permutations of $\{1,2,...,n\}$ with exactly $k$ fixed points which avoid all patterns in $T \subseteq S_m$. We enumerate $S_n^k(T)$, $T \subseteq S_3$, for all $|T| \geq 2$ and $0 \leq k \leq n$.

保持$t$-stack-sortability的$321$-和$213$-避免排列之间的双射

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。

莱默码与避免(32-1)模式的排列到避免(3-21)模式的排列的逆补映射

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

In 1990 West conjectured that there are $2(3n)!/((n+1)!(2n+1)!)$ two-stack sortable permutations on $n$ letters. This conjecture was proved analytically by Zeilberger in 1992. Later, Dulucq, Gire, and Guibert gave a combinatorial proof of this conjecture. In the present paper we study generating functions for the number of two-stack sortable permutations on $n$ letters avoiding (or containing exactly once) 132 and avoiding (or containing exactly once) an arbitrary permutation $τ$ on $k$ letters. In several interesting cases this generating function can be expressed in terms of the generating f…

A probabilistic approach to consecutive pattern avoiding in permutations

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.

避免两个平坦部分有序模式的排列计数

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。