The beginning of the paper On the foundations of combinatorial theory. VI. The idea of generating function (1972) link text says that

Since Laplace discovered the remarkable correspondence between set theoretic operations and operations on formal power series, and put it to use with great success to solve a variety of combinatorial problems, generating functions (and their continuous analogues, namely, characteristic functions) have become an essential probabilistic and combinatorial technique. A unified exposition of their theory, however, is lacking in the literature. This is not surprising, in view of the fact that all too often generating functions have been considered to be simply an application of the current methods of harmonic analysis.

Would someone explain the meaning of the paragraph in detail, please? Especially, what is

"the correspondence",

characteristic functions as continuous analogues of generating functions, and

the "current methods of harmonic analysis"?

Since I'm not familiar with harmonic analysis, I also wonder if "more current" methods in this discipline might aid solving combinatorial problems.