### Abstract:

We present exponential generating function analogues to two classical identities involving the
ordinary generating function of the complete homogeneous symmetric functions. After a suitable
specialization the new identities reduce to identities involving the first and second order
Eulerian polynomials. The study of these identities led us to consider a family of symmetric
functions associated with the Stirling permutations introduced by Gessel and Stanley. In
particular, we define certain type statistics on Stirling permutations that refine the statistics of
descents, ascents and plateaux and we show that their refined versions are equidistributed
generalizing a result of Bóna.
The definition of this family of symmetric functions extends to the generality of \(r\)-Stirling
permutations or \(r\)-multipermutations. We discuss some ocurrences of these symmetric functions in
the cases of \(r=1\) and \(r=2\) .