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.

中文翻译：

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通过生成具有一维标签的树编码的排列中连续模式的渐近正态性

我们考虑从通过生成树枚举的家族中抽取的均匀随机排列。我们开发了一种新的通用技术来为这种排列中固定模式的连续出现次数建立中心极限定理。我们提出了一种在条件随机彩色游走等系列中对均匀排列进行采样的技术。在此基础上，我们推导出随机排列中连续模式的行为，研究相应随机游走中连续增量的特性。该方法适用于具有一维标记生成树（连同一些技术假设）的排列族，并暗示此类族中随机排列的局部收敛。我们展示了十种不同的排列系列，其中大部分是排列类，满足我们的假设。据我们所知，这是第一个使用生成树来建立概率结果的工作——它被引入来枚举组合对象。