论文标题
连续模式的渐近正态性在编码的置换中通过具有一维标签的树木编码
Asymptotic normality of consecutive patterns in permutations encoded by generating trees with one-dimensional labels
论文作者
论文摘要
我们考虑从通过生成树木列举的家庭中得出的统一随机排列。我们开发了一种新的通用技术，以建立一个中心限制定理，以用于此类排列中固定模式的连续发生数量。 我们提出了一种在条件随机的彩色步道等家庭中采样均匀排列的技术。在此基础上，我们在随机排列中得出了连续模式的行为，以研究相应随机步行中连续增量的性能。该方法适用于带有一维生成树的置换家族（以及一些技术假设），这意味着此类家庭中随机排列的局部收敛。我们展示了十个不同的排列家庭，其中大多数是置换类别，可以满足我们的假设。 据我们所知，这是第一批生成树（用于列举组合物体）的生成树的作品已被用来建立概率结果。
We consider uniform random permutations drawn from a family enumerated through generating trees. We develop a new general technique to establish a central limit theorem for the number of consecutive occurrences of a fixed pattern in such permutations. We propose a technique to sample uniform permutations in such families as conditioned random colored walks. Building on that, we derive the behavior of the consecutive patterns in random permutations studying properties of the consecutive increments in the corresponding random walks. The method applies to families of permutations with a one-dimensional-labeled generating tree (together with some technical assumptions) and implies local convergence for random permutations in such families. We exhibit ten different families of permutations, most of them being permutation classes, that satisfy our assumptions. To the best of our knowledge, this is the first work where generating trees - which were introduced to enumerate combinatorial objects - have been used to establish probabilistic results.
