I know the combinatorial interpretation of first, and second order Stirling numbers (#of k cycles of n items, and #of partitions n items into k subsets). Is there an interpretation for the generalized Stirling numbers?
A combinatorial interpretation of the earlier studied generalized Stirling numbers, emerging in a normal ordering problem and its inversion, is given. It involves unordered forests of certain types of labeled trees. Partition number arrays related to such forests are also presented.
Similar numbers were introduced in the article On a generalization of Stirling numbers (2002). They have combinatorial interpretations in terms of trees and necklaces.