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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?

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Combinatorial Interpretation of Generalized Stirling Numbers (2009)

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.

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Similar numbers were introduced in the article On a generalization of Stirling numbers (2002). They have combinatorial interpretations in terms of trees and necklaces.

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