发表机构
Northwestern Polytechnical University; Shenzhen Research Institute of Northwestern Polytechnical University; University of Strathclyde（西北工业大学; 西北工业大学深圳研究院; 思克莱德大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究可分离排列中长度至多2的网格模式分布，确定长度1的所有等价类分布，并利用对称操作将长度2的等分布对划分为31个等价类，其中14个类的枚举留作开放问题。
AI 中文摘要
本文对排列中网格模式分布的长期研究做出了贡献。我们将这些研究扩展到可分离排列，并对长度至多为2的网格模式进行了全面分析。对于长度为1的网格模式，我们确定了所有六个等价类的分布。此外，我们获得了包含众所周知的排列统计量（即严格不动点）的类别中模式的联合分布。进一步地，计算机实验表明，类型为12和21且具有相同阴影的网格模式对中，至多有124对是等分布的。利用对称操作，我们将这些对按分布划分为38个等价类。通过明确确定其中24个分布（在许多情况下，实际上是相应对的联合等分布），我们将等价类的数量减少到31个，而这正是真实的等价类数量。枚举其中14个类仍是一个开放问题。
英文摘要
This paper contributes to the long line of research on the distribution of mesh patterns in permutations. We extend these studies to separable permutations and carry out a comprehensive analysis of mesh patterns of length at most~2. For mesh patterns of length~1, we determine the distributions for all six equivalence classes. In addition, we obtain the joint distribution for the patterns in the class containing the well-known permutation statistic known as the strict fixed point. Furthermore, computer experiments suggest that at most 124 pairs of mesh patterns of type 12 and 21 with identical shading are equidistributed. Using symmetry operations, we partition these pairs into 38 equivalence classes with respect to distribution. By explicitly determining 24 of these distributions (in many cases, in fact, joint equidistributions of the respective pairs), we reduce the number of equivalence classes to~31, which turns out to be the true number of equivalence classes. Enumerating 14 of these classes is left as an open problem.
CommentsAccepted for publication in Discrete Applied Mathematics
