避免模式的排列回避游戏与反向回应

Permutation groups, pattern involvement, and Galois connections

There is a connection between permutation groups and permutation patterns: for any subgroup $G$ of the symmetric group $S_\ell$ and for any $n \geq \ell$, the set of $n$-permutations involving only members of $G$ as $\ell$-patterns is a subgroup of $S_n$. Making use of the monotone Galois connection induced by the pattern avoidance relation, we characterize the permutation groups that arise via pattern avoidance as automorphism groups of relations of a certain special form. We also investigate a related monotone Galois connection for permutation groups and describe its closed sets and kernels …

Bounded affine permutations II. Avoidance of decreasing patterns

We continue our study of a new boundedness condition for affine permutations, motivated by the fruitful concept of periodic boundary conditions in statistical physics. We focus on bounded affine permutations of size $N$ that avoid the monotone decreasing pattern of fixed size $m$. We prove that the number of such permutations is asymptotically equal to $(m-1)^{2N} N^{(m-2)/2}$ times an explicit constant as $N\to\infty$. For instance, the number of bounded affine permutations of size $N$ that avoid $321$ is asymptotically equal to $4^N (N/4π)^{1/2}$. We also prove a permuton-like result for the…

有界仿射排列 I. 模式回避与计数

Bounded affine permutations I. Pattern avoidance and enumeration

本文引入仿射排列的有界性条件，研究模式回避与计数，证明回避特定递增振荡模式蕴含该条件，并利用解析方法关联块分解排列与普通排列的计数，完成有界仿射排列的精确与渐近计数。

交错符号矩阵的 key 规避

Key-avoidance for alternating sign matrices

本文系统研究 ASM 的 key 模式规避，给出多类枚举、与无间隙单调三角形的双射，并借助 Schubert 多项式得到 Catalan 恒等式。

具有模式受限循环形式的模式受限循环排列

Pattern-restricted cyclic permutations with a pattern-restricted cycle form

本文研究了避免单调递减排列的循环排列，在循环标准形式避免特定长度模式或所有循环旋转避免特定长度模式的情况下，分别通过连分数生成函数给出答案并枚举了相关情形。

关于对称模式回避集

On symmetric pattern avoidance sets

本文研究了对称模式回避集的性质，证明了对称回避集在特定条件下具有Schur正性，并解决了关于对称集合大小的分类问题。

长度为3的置换的Shape-Wilf-序

Shape-Wilf-ordering of permutations of length 3

迄今为止，关于模式回避的研究对排列的Wilf序的认识有限。[001 pdf 1st-31 files] 1998年12月20日，美国加州大学洛杉矶分校（|S_n（τ）|）和进一步的工作建议分层与单调模式的渐近排序。然而，博纳提供了迄今为止唯一已知的结果，其类型的排序排列：|S_n（1342）|<|中文（简体）|<|S_n（1324）|n>6。我们通过将S_3排序到更强的形状Wilf序来给出这个结果的不同证明：|中文（简体）|<=|中文（简体）|<=|中文（简体）|对于任何Young图Y，作为推论，|S_Y（k+2，k+1，k+3，τ）|<=| S_Y（k+1，k+2，k+3，τ）|<= |S_Y（k+3，k+1，k+2，τ）|对于S_k中的任何τ，并找出何时得到等式。(In特别的，对于特定的Y，我们发现|中文（简体）|=|中文（简体）|与所有其他Fibonacci项一致。这加强和推广博纳的结果任意长度的排列。虽然所有长度为3的排列都以多种方式被证明是Wilf等价的，但本文通过使用所有的Young图来区分和排序这些排列。这就引出了一个问题：排列的形状-威尔夫序，或者它的某种推广，是否不是接近模式避免序的“真正”方式。

Combinatorial generation via permutation languages. I. Fundamentals

In this work we present a general and versatile algorithmic framework for exhaustively generating a large variety of different combinatorial objects, based on encoding them as permutations. This approach provides a unified view on many known results and allows us to prove many new ones. In particular, we obtain four classical Gray codes for permutations, bitstrings, binary trees and set partitions as special cases. We present two distinct applications for our new framework: The first main application is the generation of pattern-avoiding permutations, yielding new Gray codes for different fami…