This is the first of a sequence of research reports that attempt to approach the topic of the algebra of set functions in a complete way, and to illustrate its practicality in (re-)defining and (re-)proving results that we might have missed in the context of counting matches, colorings, or acyclic orientations. Here, Lass introduces the algebra and its generating functions. Theorems relative to the enumeration of a variety of functions are presented using this framework.

In the tradition of enumerative combinatorics, ordinary and exponential generating functions are used. General product and weighted product theorems are stated and proved. The paper closes with a general duality theorem. In the ample examples used to illustrate the results, the author considers enumerations of different classes of functions. The examples draw on a broad range of subjects, including injections, surjections, and matching and hypergraph colorings.

The presented approach is encouraging. It is comprehensive in the sense that the author explores the framework systematically, and uses familiar, frequently used classes of functions from enumeration problems. I hope that in the successors to this paper, we will continue to see familiar results presented in a different framework, with proofs that will inspire new approaches and algorithms for dealing with familiar results and knowledge in an easier way.