Decreasing Runs in Quasi-Stirling Permutations of Multisets

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.