Generalized Barred Preferential Arrangements

We investigate a generalization of Fubini numbers. We present the combinatorial interpretation as barred preferential arrangements with some additional conditions on the blocks. We provide a proof for a generalization of Nelsen's Theorem. We consider these numbers from a probabilistic view point and demonstrate how they can be written in terms of the expectation of random descending factorial involving the negative binomial process.

Keywords:

barred preferential arrangements , generalized Stirling numbers , geometric polynomials , negative binomial process