  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.
