AI 中文总结
本文证明多重集拟斯特林排列的降序游程具有分布不变性，构造保留降序游程的多重度重分配双射，结合经典游程定理导出相关统计量的递推关系与生成函数。
AI 中文摘要
拟斯特林排列是斯特林排列的自然扩展，属于多重排列π，满足对任意满足π_j₁=π_j₃且π_j₂=π_j₄的子序列π_j₁π_j₂π_j₃π_j₄，有π_j₁=π_j₂。Yan、Yang、Huang和Zhu利用双射构造证明，多重集M={1^k₁,2^k₂,…,n^kₙ}的拟斯特林排列的上升、下降和平台的联合分布，与多重集M'={1^(k₁+…+kₙ−n+1),2,…,n}上的对应分布一致。本文证明降序游程也具有相同的分布不变性，进而所有降序连续模式均满足该性质。为此，本文遵循Yan-Yang-Huang-Zhu的方法，构造了一种保留降序游程的多重度重分配双射，将M上拟斯特林排列的降序连续模式联合分布计算归约至M'上。结合经典游程定理，该双射可导出这些统计量分布函数的显式递推关系和生成函数。
英文摘要
As a natural extension of Stirling permutations, quasi-Stirling permutations are multipermutations $π$ with the property that for any subsequence $π_{j_1}π_{j_2}π_{j_3}π_{j_4}$ satisfying $π_{j_1}=π_{j_3}$ and $π_{j_2}=π_{j_4}$, we have $π_{j_1}=π_{j_2}$. Using a bijective construction, Yan, Yang, Huang and Zhu showed that the joint distribution of ascents, descents and plateaux over quasi-Stirling permutations of a multiset $M=\{1^{k_1},2^{k_2},\ldots,n^{k_n}\}$ coincides with that over the multiset $M'=\{1^{k_1+\cdots+k_n-n+1},2,\ldots,n\}$. In this paper, we prove that the same invariance of distribution holds for decreasing runs, and consequently for all decreasing consecutive patterns. To this end, following the Yan-Yang-Huang-Zhu approach, we construct a multiplicity-redistribution bijection that preserves decreasing runs, thereby reducing the computation of joint distribution of decreasing consecutive patterns over quasi-Stirling permutations from $M$ to $M'$. Together with the classical run theorem, our bijection leads to explicit recurrence relations and generating functions for the distribution functions of these statistics over quasi-Stirling permutations.
