Counting Stirling permutations by number of pushes

Let 𝒯^(k)_n denote the set of k-Stirling permutations having n distinct letters. Here, we consider the number of steps required (i.e., pushes) to rearrange the letters of a member of 𝒯^(k)_n so that they occur in non-decreasing order. We find recurrences for the joint distribution on 𝒯^(k)_n for the statistics recording the number of levels (i.e., occurrences of equal adjacent letters) and pushes. When k = 2, an explicit formula for the ordinary generating function of this distribution is also found. In order to do so, we determine the LU-decomposition of a certain infinite matrix having polynomial entries which enables one to compute explicitly the inverse matrix.