  A permutation $\sigma$ of a multiset is called Stirling permutation if
$\sigma(s)\ge \sigma(i)$ as soon as $\sigma(i)=\sigma(j)$ and $i<s<j.$ In our
paper we study Stirling polynomials that arise in the generating function for
descent statistics on Stirling permutations of any multiset. We develop
generalizations of the classical Stirling numbers and present their
combinatorial interpretations. Particularly, we apply the theory of
$P$-partitions. Using certain specifications we also introduce the Stirling
numbers of odd type and generalizations of the central factorial numbers.
