  We introduce and study a new notion of patterns in Stirling and $k$-Stirling
permutations, which we call block patterns. We prove a general result which
allows us to compute generating functions for the occurrences of various block
patterns in terms of generating functions for the occurrences of patterns in
permutations. This result yields a number of applications involving, among
other things, Wilf equivalence of block patterns and a new interpretation of
Bessel polynomials. We also show how to interpret our results for a certain
class of labeled trees, which are in bijection with Stirling permutations.
