The k-Stirling permutations are a generalization of both ordinary permutations and Stirling permutations introduced by Gessel and Stanley. In this paper, we introduce the statistics “linv” and “lmaj” on k-Stirling permutations and prove that they are equidistributed. This generalizes the famous result of MacMahon that the permutation statistics inversion number and major index are equidistributed. Some further MacMahon type results for k-Stirling permutations are also given. As an application, we obtain a “peak-based” Mahonian permutation statistic.

Moreover, we find three equidistributed bi-statistics on k-Stirling permutations, and call their common generating function the $1/k$-Euler-Mahonian polynomial, which unifies the Eulerian, $1/k$-Eulerian and Euler-Mahonian polynomials. The $1/k$-Euler-Mahonian polynomial is a q-analogue of the $1/k$-Eulerian polynomial, this solves a problem posed by Savage and Viswanathan.

所述ķ -Stirling排列是通过Gessel和Stanley引入两个普通的排列和斯特林排列的概括。在本文中，我们介绍了有关k-斯特林排列的统计量“ linv”和“ lmaj”，并证明了它们是均匀分布的。这概括了MacMahon的著名结果，即排列统计量的反转数和主要指标是均匀分布的。还给出了k-斯特林置换的一些进一步的MacMahon类型结果。作为一个应用程序，我们获得了“基于峰值”的Mahonian置换统计量。

此外，我们在k-斯特林排列上找到了三个等距的双统计量，并将它们的共同生成函数称为$\mathrm{1个}/ķ$-Euler-Mahonian多项式，它统一了Eulerian，$\mathrm{1个}/ķ$-Eulerian和Euler-Mahonian多项式。这$\mathrm{1个}/ķ$-Euler-Mahonian多项式是q的-analogue$\mathrm{1个}/ķ$-欧拉多项式，这解决了Savage和Viswanathan提出的问题。