Cycle index series of structures over digraphs (Q1586766)

Beginning with an overview of species on digraphs imbedded into the category of coloured species, the author introduces \(Z\)-series for species on digraphs. A logarithmic connection between multiset permutations and necklasses has also been established. Using some of the \(Z\)-series computed in this paper, the author has obtained classical and new generating functions for unlabeled structures over digraphs. Of particular interest is the generating function for multiset permutations, which is a matrix-of-variables generalization of Euler's classical generating function for the number of partitions of nonnegative integers.