Wilf classification of three and four letter signed patterns
W. M. B. Dukes, T. Mansour, A. Reifegerste
我们给出了一些新的Wilf等价的签署模式，使长度为3和4的签署模式的完整分类。该问题被认为是一般的，以及对合符号置换的模式避免。
We give some new Wilf equivalences for signed patterns which allow the complete classification of signed patterns of lengths three and four. The problem is considered for pattern avoidance by general as well as involutive signed permutations.
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Restricted single or double signed patterns
令$E_n^r=\{[τ]_a=(τ_1^{(a_1)},...,τ_n^{(a_n)})| τ\in S_n,\ 1\leq a_i\leq r\}$是符号1，2，.上的所有有符号置换的集合，n的符号为1，2，...，R.我们证明了，对于每个2-字母符号模式$[τ]_a$，$E_n^r$中避免$[τ]_a$的符号置换的数目由公式$\sum\limits_{j=0}^n j!(r-1)^j{n\choose j}^2$给出。此外，我们证明了只有一个Wilf类r=1，四个Wilf类r=2，和六个Wilf类$r\geq 3$。
Recently, it has been determined that there are 242 Wilf classes of triples of 4-letter permutation patterns by showing that there are 32 non-singleton Wilf classes. Moreover, the generating function for each triple lying in a non-singleton Wilf class has been explicitly determined. In this paper, toward the goal of enumerating avoiders for the singleton Wilf classes, we obtain the generating function for all but one of the triples containing 1324. (The exceptional triple is conjectured to be intractable.) Our methods are both combinatorial and analytic, including generating trees, recurrence …
We determine all 242 Wilf classes of triples of 4-letter patterns by showing that there are 32 non-singleton Wilf classes. There are 317 symmetry classes of triples of 4-letter patterns and after computer calculation of initial terms, the problem reduces to showing that counting sequences that appear to be the same (agree in the first 16 terms) are in fact identical. The insertion encoding algorithm (INSENC) accounts for many of these and some others have been previously counted; in this paper, we find the generating function for each of the remaining 36 triples and it turns out to be algebrai…
This paper is one of a series whose goal is to enumerate the avoiders, in the sense of classical pattern avoidance, for each triple of 4-letter patterns. There are 317 symmetry classes of triples of 4-letter patterns, avoiders of 267 of which have already been enumerated. Here we enumerate avoiders for all small Wilf classes that have a representative triple containing the pattern 1342, giving 40 new enumerations and leaving only 13 classes still to be enumerated. In all but one case, we obtain an explicit algebraic generating function that is rational or of degree 2. The remaining one is show…
Super-strong (elsewhere referred to as strong) Wilf equivalence is a type of Wilf equivalence on words that was introduced by Kitaev et al. in 2009. We provide a necessary and sufficient condition for two permutations in $n$ letters to be super-strongly Wilf equivalent, using distances between letters within a permutation. Furthermore, we give a characterization of such equivalence classes via two-colored binary trees. This allows us to prove, in the case of super-strong Wilf equivalence, the conjecture stated in (Kitaev et al., 2009) that the cardinality of each Wilf equivalence class is a po…
New equivalences for pattern avoiding involutions
我们完成了长度为5的符号模式的Wilf分类，包括符号置换和符号对合。给出了模式的新的一般等价关系，证明了Jaggard关于对称群中对合的命题，避免了某些长度为5和6的模式.用这种方法，我们还完成了S_5，S_6和S_7的排列和对合的Wilf分类。
Stanley-Wilf limits are typically exponential
对于排列$π$，令$S_{n}(π)$为$n$字母上避免$π$的排列数。Marcus和Tardos证明了著名的Stanley-Wilf猜想：$L(π)= \lim_{n \to \infty} S_n(π)^{1/n}$存在且有限。多年来，许多研究者在数值证据的支持下证明，$k$字母上的每个排列$π$都有$L(π)=Θ(k^2)$。我们反驳了这一猜想，表明$L(π)=2^{k^{Θ(1)}}$几乎所有的排列$π$上$k$字母。
Avoiding 2-letter signed patterns
设B_n是超八面体群，即n个字母上所有符号置换的集合，B_n（T）是B_n中避开符号模式集合T的所有符号置换的集合。本文给出了集合B_n（T）的所有基数，其中$T \subseteq B_2$ .这允许我们通过二项式系数、二项式系数、卡塔兰数和斐波那契数的逆来表示这些基数。
We generalize three classical characterizations of line graphs to line graphs of signed and gain graphs: the Krausz's characterization, the van Rooij and Wilf's characterization and the Beineke's characterization. In particular, we present a list of forbidden gain subgraphs characterizing the class of gain-line graphs. In the case of a signed graph whose underlying graph is a line graph, this list consists of exactly four signed graphs. Under the same hypothesis, we prove that a signed graph is the line graph of a signed graph if and only if its eigenvalues are either greater than $-2$, or les…
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。
Proof of a conjecture on the shape-Wilf-equivalence for partially ordered patterns
本文通过在给定 Young diagram 的两组避模式横截之间建立双射，证实了 Burstein-Han-Kitaev-Zhang 提出的关于长度为 $k$ 的偏序模式（POP）的 shape-Wilf-equivalence 猜想。
Enumeration of super-strong Wilf equivalence classes of permutations in the generalized factor order
本文通过递归公式枚举广义因子序下排列的超强Wilf等价类，并揭示其与移位等价的联系，从而完整描述并枚举排列的移位等价类。
On avoidance of patterns of the form σ-τ by words over a finite alphabet
椭圆形或虚线模式类似于经典模式，只是出现的某些字母需要相邻。我们证明了几个无限家庭的威尔夫等价的k元的话，涉及包含一个破折号，这解释了大多数的长度为4的这种模式的等价见证的vincular模式。当结合以前的结果，数值证据，并在特定情况下的一些参数，我们得到完整的威尔夫分类长度为4的所有vincular模式包含一个破折号。在某些情况下，我们的证明进一步表明，等价性保持多集排列，因为它被看作是尊重每个字母在一个单词中出现的次数。对长度为4的模式σ给出了一些相关的计数结果，其中包括关于[k]^n的成员个数的生成函数公式，避免了形式为11 a-b的任何σ.
Generating functions for Wilf equivalence under generalized factor order
Kitaev、Liese、Remmel和Sagan最近定义了由偏序集合$(P, \leq_P)$中的字母组成的单词的广义因子序，如果存在与$u$相同长度的$w$的子字$v$，则设置$u \leq_P w$，使得对于所有$i$，$v$的第$i$ -个字符大于或等于$u$的第$i$ -个字符。这个子字$v$被称为$u$到$w$的嵌入。对于$P$是具有通常顺序的正整数的情况，他们将单词$w = w_1\ldots w_n$的权重定义为$\text{wt}(w) = x^{\sum_{i=1}^n w_i} t^{n}$，以及相应的权重生成函数$F(u;t,x) = \sum_{w \geq_P u} \text{wt}(w)$。然后，他们定义了两个词$u$和$v$是Wilf等价的，记为$u \backsim v$，当且仅当$F(u;t,x) = F(v;t,x)$。他们还定义了相关的生成函数$S(u;t,x) = \sum_{w \in \mathcal{S}(u)} \text{wt}(w)$，其中$\mathcal{S}(u)$是所有词$w$的集合，使得$u$到$w$中的唯一嵌入是$w$的后缀，并证明了$u \backsim v$当且仅当$S(u;t,x) = S(v;t,x)$。我们继续这项研究，给出了一个明确的公式$S(u;t,x)$，如果$u$因素成一个弱增加的字，其次是一个弱…
We consider avoidance of permutation patterns with designated gap sizes between pairs of consecutive letters. We call the patterns having such constraints distant patterns (DPs) and we show their relation to other pattern notions investigated in the past. New results on DPs with 2 and 3 letters are obtained. Furthermore, we show how one can use DPs to prove two former conjectures of Kuszmaul without a computer. In addition, we deduce a surprising relation between the sets of permutations avoiding the classical patterns $123$ and $132$ by looking at a class of DPs with tight gap constraints. So…
In this paper, we study the Wilf-type equivalence relations among multiset permutations. We identify all multiset equivalences among pairs of patterns consisting of a pattern of length three and another pattern of length at most four. To establish our results, we make use of a variety of techniques, including Ferrers-equivalence arguments, sorting by minimal/maximal letters, analysis of active sites and direct bijections. In several cases, our arguments may be extended to prove multiset equivalences for infinite families of pattern pairs. Our results apply equally well to the Wilf-type classif…
Pattern-avoiding Cayley permutations via combinatorial species
本文采用组合物种方法系统研究Cayley排列的模式避免问题，导出了长度至多为3的模式的物种方程、生成级数与计数公式，并推广了无平步结构，探讨了Wilf等价概念。
Cyclic permutations realized by signed shifts
映射的周期（有序）模式是由其周期轨道中的点的相对顺序实现的排列。利用与任意符号移位相关的某个轮换的下降集的结构，给出了该移位的周期模式的组合刻画。符号移位是一类重要的一维动力系统，它包括移位映射和帐篷映射。定义为有限字母表上的无限单词集合上的函数，有符号移位删除第一个字母，并且根据其值，可能对剩余的单词应用求补操作。对于移位映射、逆移位映射和帐篷映射，我们给出了它们的周期模式数的精确公式。作为我们的工作的副产品，我们恢复Gessel-Schottenauer和Weiss-Rogers的结果，并获得新的计数公式的模式避免循环。
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