By Basic Combinatorics

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Nk−1 , n − n1 − · · · − nk−1 1 2 ✂ 38 Chapter 7 Combinatorics and Probability Enumerative combinatorics provides one of the basic foundations of probability theory. For if S is a finite set with subset E and one chooses an element of S at random, the probability that the element chosen actually belongs to E is given by the formula |E|/|S|. The determination of the cardinalities of various sets thus underlies this probability model (the so-called “uniform model”) in a crucial way. The applications of enumerative combinatorics are by no means limited to uniform models.

Uk . Instead of distributing balls, we distribute the elements of A, whatever they may be. And instead of placing these elements in actual urns, we enclose them in curly brackets, and arrange the resulting sets in a sequence. 53 CHAPTER 9. SURJECTIONS AND ORDERED PARTITIONS With σ(n, k) viewed as the number of surjections from an n-set to a k-set, it isn’t obvious that the row sum n Pn := σ(n, k) k=0 is of any combinatorial interest. But now that we have an alternative interpretation of σ(n, k) as the number of ordered partitions of an n-set with k blocks, it is clear that Pn is the total number of ordered partitions of an n-set with all possible numbers of blocks.

Ii) The identity relation, =, on S is defined extensionally as the subset {(x, x) : x ∈ S} of S × S, and intensionally by stipulating that x = y if and only if x and y are the same element of S. (iii) The universal relation on S is the subset of S × S consisting of S × S itself. The relation ρ on S is the universal relation on S if and only if xρy for all x, y ∈ S. For example, if S is a set of real numbers and one defines xρy to mean that x < y or x ≥ y, then ρ is the universal relation on S.