A family of symmetric functions associated with Stirling permutations

R. S. González D'León

Arxiv 1506.01628.

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 \(r=2\) .