Stirling permutations were introduced by Gessel and Stanley in 1978, who enumerated them by the number of descents to give a combinatorial interpretation of certain polynomials related to Stirling numbers. A natural extension of these permutations are quasi-Stirling permutations, which can be viewed as labeled noncrossing matchings. They were recently introduced by Archer et al., motivated by the fact that the Koganov–Janson correspondence between Stirling permutations and labeled increasing plane trees extends to a bijection between quasi-Stirling permutations and the same set of trees without the increasing restriction.

In this paper we prove a conjecture of Archer et al. stating that there are ${(n+1)}^{n-1}$ quasi-Stirling permutations of size n having n descents. More generally, we give the generating function for quasi-Stirling permutations by the number of descents, expressed as a compositional inverse of the generating function of Eulerian polynomials. We also find the analogue for quasi-Stirling permutations of the main result from Gessel and Stanley's paper. We prove that the distribution of descents on these permutations is asymptotically normal, and that the roots of the corresponding quasi-Stirling polynomials are all real, in analogy to Bóna's results for Stirling permutations.

Finally, we generalize our results to a one-parameter family of permutations that extends k-Stirling permutations, and we refine them by also keeping track of the number of ascents and the number of plateaus.

斯特林排列是由Gessel和Stanley在1978年提出的，他们根据下降的次数对它们进行了枚举，从而对与斯特林数有关的某些多项式进行了组合解释。这些排列的自然扩展是准斯特林排列，可以将其视为标记的非交叉匹配。它们是由Archer等人最近引入的，其动机是斯特林排列与标记的增加的平面树之间的Koganov-Janson对应关系扩展到准斯特林排列与同一树集之间的双射而没有增加的限制。

在本文中，我们证明了Archer等人的猜想。指出有${（ñ+\mathrm{1个}）}^{ñ-\mathrm{1个}}$准斯特林大小的置换Ñ具有Ñ下坡。更一般地，我们通过下降的数量给出准斯特林置换的生成函数，表示为欧拉多项式生成函数的组成逆。我们还从Gessel和Stanley的论文中找到了主要结果的准斯特林置换的类似物。我们证明了在这些排列上的下降分布是渐近正态的，并且类似于Bóna的斯特林排列结果，相应的准斯特林多项式的根都是实数。

最后，我们将结果推广到一个扩展了k -Stirling排列的单参数排列族，并且通过跟踪上升次数和高原次数来完善它们。