How to Count

The basic problem of enumerative combinatorics is that of counting the number of elements of a finite set. Usually we are given an infinite class of finite sets Si where i ranges over some index set I (such as the nonnegative integers ℕ), and we wish to count the number f(i) of elements of each Si “simultaneously.” Immediate philosophical difficulties arise. What does it mean to “count” the number of elements of Si? There is no definitive answer to this question. Only through experience does one develop an idea of what is meant by a “determination” of a counting function f(i). The counting function f(i) can be given in several standard ways:

1. The most satisfactory form of f(i) is a completely explicit closed formula involving only well-known functions, and free from summation symbols. Only in rare cases will such a formula exist. As formulas for f(i) become more complicated, our willingness to accept them as “determinations” of f(i) decreases. Consider the following examples.

1.1.1. Example. For each n ∈ ℕ, let f(n) be the number of subsets of the set [n] = {1, 2, …, n}. Then f(n) = 2n, and no one will quarrel about this being a satisfactory formula for f(n).

1.1.2. Example. Suppose n men give their n hats to a hat-check person. Let f(n) be the number of ways that the hats can be given back to the men, each man receiving one hat, so that no man receives his own hat.