Counting Functions

Consider the family of n-subsets of a set R or the family of r-partitions of a set N. In chapter I, these families were viewed as special patterns whereas in chapter II they appeared as levels of certain lattices. In this chapter we want to count patterns and lattice levels starting from table 1.13 on the one hand and the fundamental examples of section II.4 on the other. In accordance with the set-up of the book we shall concentrate more on deriving general counting principles rather than on supplying a large set of recursion and inversion formulae for well-known coefficients such as the binomial coefficients or various partition numbers. For a good collection of the latter the reader is referred to Riordan [1, 2] or to Knuth [1. vol. 1].