The Structure of Permutation Classes

这个项目的总体目标是扩展排列类的结构理论，统一该理论中不同的分支，并通过将新兴观点和工具箱应用于精确枚举和增长率表征问题来测试它们的力量。历史上，排列类的研究起源于20世纪60年代和70年代两个独立的流派。一个本质上是组合的，专注于短（长度最多为4）禁止模式的小集合（通常为1或2）的排列枚举问题。另一个来自理论计算机科学，关注的是由普通排序机制和它们的组合产生的集合。在过去10年左右的时间里，这两条线走得更近了，这种相互作用创造了一个新的、快速发展的组合学领域，与理论计算机科学、可计算性和复杂性理论、代数和计算生物学等有重要的相互作用。除了对排序机制和枚举问题的持续兴趣外，还出现了主要的新研究方向，包括类的结构理论、类的渐近行为、广义模式回避、填充密度、算法和可判定性问题以及几何方法。

The overarching aim of this project is to expand the structural theory of permutation classes, to unify the disparate strands coming together in this theory, and to test the power of the emerging viewpoint and toolbox by bringing them to bear on problems of exact enumeration and the characterization of growth rates.Historically, the study of permutation classes arose from two independent streams in 1960s and 1970s. One was combinatorial in nature, and concentrated on the enumeration problems for permutations with a small set (size 1 or 2, typically) of short (length up to 4) forbidden patterns. The other was coming from Theoretical Computer Science, and was concerned with sets arising from common sorting mechanisms and their combinations. In the past 10 years or so these two strands have come much closer together, and this interaction has created a new, fast developing area of combinatorics, with significant interactions with Theoretical Computer Science, the Theory of Computability and Complexity, Algebra, and Computational Biology, to name only a few. Apart from the continued interest in sorting mechanisms and enumeration problems, major new strands of research have emerged including the structural theory of classes, the asymptotic behavior of classes, generalized pattern avoidance, packing densities, algorithmic and decidability problems, and geometrical methods.