Enumeration of flattened $k$-Stirling permutations with respect to descents

A $k$-Stirling permutation of order $n$ is said to be “flattened” if the leading terms of its increasing runs are in ascending order. We show that flattened $k$-Stirling permutations of order $n+1$ are in bijection correspondence with a colored variant of type $B$ set partitions of $[-n,n]$, introduced by D.G.L. Wang. Using the theory of weighted labelled structures, we give the exponential generating functions of their cardinality and their descent enumerating polynomials. We also provide enumerative formulae for the number of flattened $k$-Stirling permutations of order $n$ with small number of descents and the number of flattened Stirling permutations with maximum number of descents.