Some statistics on Stirling permutations and Stirling derangements

$1/k$-Eulerian Polynomials and $k$-Inversion Sequences

“…In [8], Ma and Yeh provided a constructive proof that the number of ascent plateaus of 2-Stirling permutations of order n is equidistributed with a weighted variant of the number of excedances in permutations of length n, where the weight is 2 n−cyc (π) . Very recently, Duh et al [5,Lemma 8] established a bijection between 2-colored permutations and Stirling permutations. Expanding [5,Lemma 8] and combining Lemma 5 and Lemma 8, in this section, we will present a bijective proof that the ascent plateau number over k-Stirling permutations of order n is equidistributed with the ascent number over k-inversion sequences of length n.…”

“…Very recently, Duh et al [5,Lemma 8] established a bijection between 2-colored permutations and Stirling permutations. Expanding [5,Lemma 8] and combining Lemma 5 and Lemma 8, in this section, we will present a bijective proof that the ascent plateau number over k-Stirling permutations of order n is equidistributed with the ascent number over k-inversion sequences of length n.…”

“…For some special cases of s, the s-inversion Eulerian polynomial has been extensively studied. For example, for s = (1,4,3,8,5,12, . .…”

Counting Stirling permutations by number of pushes

“…When k = 2, these are the Stirling permutations originally considered by Gessel and Stanley [3]. The k-Stirling permutations have been provided various refined counts in the case k = 2 (see, e.g., [1,2,5,6]) and for general k (see [7]). It is seen that there are n−1 i=1 (ki + 1) k-Stirling permutations of order n. In this paper, we consider a new refinement of this number and in particular, when k = 2, a new q-analogue of the odd double factorial sequence (see, e.g., [10, A001147]).…”