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11. Introduction to Exponential Generating Functions

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11. Introduction to Exponential Generating Functions 11. Introduction to Exponential Generating Functions. We have seen several applications of generating functions – more specifically, of ordinary generating functions. Exponential generating functions are of another kind and are useful for solving problems to which ordinary generating functions are not applicable. Ordinary generating functions arise when we have a (finite or countably in- finite) set of objects S and a weight function ω : S → Nr. Then the ordinary ω generating function ΦS(x) is defined and we can proceed with calculations. Expo- nential generating functions arise in a somewhat more complicated situation. The basic idea is that they are used to enumerate “combinatorial structures on finite sets”. In this section I will try to give you some idea of what this means without getting bogged down in an axiomatic development. In the process, we will be able to derive enough of the theory to solve some interesting problems. A more formal treatment of the subject is postponed until Chapter 12, which includes proper foun- dations of the theory as well as discussion of some subtle issues which we can’t even speak about until the language is developed. But all that is for later! Right now, let’s concentrate on the general ideas, and save the niggling for when we’ve already got the big picture. So – a “combinatorial structure on a finite set” – just what does that mean? Graphs are good examples: a graph G = (V, E) consists of a finite set V together with some additional structure, in this case a set E of two–element subsets of V . Endofunctions are also good examples: an endofunction is a finite set V together with some additional structure, in this case a function φ : V → V from V to itself. Generally, a “combinatorial structure on a finite set” means a finite set together with some additional information defined in terms of that set. Of course, we are interested in counting things. So we will not consider just one (combinatorial) structure (on a finite set), but an entire family of related structures, called a class of structures. For example, we can consider the class G of all graphs. This consists of all finite sets V and all graph structures G = (V, E) on these finite sets. It’s a pretty big thing! (In fact, it is way too big even to be a set – but that’s another story.) The point about the class G is that to each finite set X it associates the finite set GX of all graphs which have X as their set of vertices. Notice that if X 6= Y are two different finite sets then GX ∩ GY = ∅, since if G ∈ GX ∩ GY then n(n−1)/2 X = V (G) = Y . Also notice that if #X = n then #GX = 2 , so that #GX depends only on #X. These properties are the essential ones that we abstract to define a class of structures. Definition 11.1 (Classes of Structures). A class A of structures associates to every finite set X another finite set AX , in such a way that the following two conditions are satisfied: (i) if X 6= Y are distinct finite sets then AX ∩ AY = ∅; (ii) if X and Y are finite sets with #X = #Y , then #AX = #AY . (In Section 12 we will enrich this definition and speak about “natural” classes, but this will suffice for now.) The interpretation of the class A is that AX is the finite set of combinatorial structures in the class A which are defined in terms of the finite set X of “vertices”. Now we see the kind of enumeration problem that exponential generating functions are designed to solve: given a class A of structures, determine #AX for all finite sets X. That is, determine how many A–structures are defined on each finite set. Of course, by condition (ii) of Definition 11.1, this amounts to determining #ANn for all n ∈ N. The notation ANn is a bit awkward, so let’s use An := ANn to mean the same thing. We could put all the numbers #An for n ∈ N together into a generating function: ∞ X n (#An)x n=0 but it turns out that this is not the way to do it. The reason why this is no good is that the combinatorial operations we will use to analyze and manipulate classes of structures are not reflected by algebraic operations on these power series. Instead, the proper way to translate the combinatorics into algebra in this situation is as follows. Definition 11.2 (Exponential Generating Functions). Let A be a class of struc- tures. The exponential generating function of A is ∞ X xn A(x) := (#A ) . n n! n=0 (This definition will be embellished a little later to include more indeterminates, but this is the essential form.) Let’s illustrate this with a few cheap examples for which we already know the answer. Example 11.3. First, consider the class S of permutations: to each finite set X it associates the finite set SX of all bijections σ : X → X from X to X. Condition (i) is easy, and condition (ii) follows from Example 2.2, so that S satisfies Definition 11.1. Way back in Theorem 2.1 we saw that #Sn = n! for all n ∈ N, so that the exponential generating function for the class of permutations is ∞ ∞ X xn X xn 1 S(x) := (#S ) = n! = . n n! n! 1 − x n=0 n=0 Example 11.4. Second, consider the class C of cyclic permutations. Recall from Exercise 2.2 that #C0 = 0 and #Cn = (n − 1)! for all n ≥ 1. Condition (i) is easy, and you should think about how to verify condition (ii). Therefore, C is a class and its exponential generating function is ∞ ∞ X xn X xn 1 C(x) := (#C ) = (n − 1)! = log n n! n! 1 − x n=0 n=1 by Example 7.9(a). Example 11.5. Third, consider the class E of finite sets: to each finite set X this associates the set EX := {X} containing only X. This seems like much ado about nothing, but it will be very useful in a little while. That is, E is the class of finite sets with no additional structure. Conditions (i) and (ii) are clear in this case. The exponential generating function for the class E is ∞ ∞ X xn X xn E(x) := (#E ) = = exp(x) n n! n! n=0 n=0 by Example 7.9(b). Notice that we have the relation 1 1 = exp log 1 − x 1 − x among these power series. Using the names of the exponential generating functions, that is S(x) = E(C(x)). This suggests that some combinatorial relation exists among the classes S, C, and E – a relation which it would be sensible to denote by something like S ≡ E[C]. In fact this is the case – a permutation is equivalent to a finite set of pairwise disjoint cyclic permutations. Our first task is to develop enough of the theory to make sense of an expression like S ≡ E[C]. The usefulness of this theory stems from the ability to identify combinatorial relations among classes – as above – and then to translate these into functional equations for the corresponding exponential generating functions. After that, one can apply algebraic techniques such as the Lagrange Implicit Function Theorem to extract the coefficients of these generating functions, thereby solving the relevant enumeration problems. To realize this program we need to discuss several opera- tions on classes of structures. Then we will have developed enough technique to analyze some nontrivial problems and derive some interesting results. (On a first pass through these constructions it might help to read the first several carefully and then skim quickly through the rest. After seeing how they are applied in a few problems one can return and reread them all carefully.) Definition 11.6 (Equivalence of Classes). Two classes A and B are (numerically) equivalent if #AX = #BX for every finite set X. Of course, this is the case if and only if their exponential generating functions are equal: A(x) = B(x). We use the notation A ≡ B to denote that A and B are equivalent. (This concept of equivalence will be superseded in Section 12 by the much more interesting concept of “natural equivalence”, but this will do for now.) Definition 11.7 (Local Finiteness and Sums of Classes). Let (A(1), A(2),...) be a (finite or infinite) sequence of classes. We say that this sequence is locally finite (i) provided that for every finite set X, at most finitely many of the sets (AX : i ≥ 1) are not empty. If this is the case then the set ∞ [ (i) BX := {i} × AX i=1 is a finite union of finite sets, so that BX is a finite set. A typical element of BX is (i) an ordered pair (i, α) with i ≥ 1 and α ∈ AX . The presence of the first coordinate (i) ensures that the sets {i} × AX are pairwise disjoint. Since the union defining BX is a disjoint union, we have ∞ X (i) #BX = #AX . i=1 Conditions (i) and (ii) of Definition 12.1 can now be verified easily for B. In sum- mary, for a locally finite sequence of classes (A(i) : i ≥ 1) the class B is well–defined, and is called the sum of the sequence. The exponential generating function of B is ∞ X B(x) = A(x). i=1 The sum of classes A(i) for i ≥ 1 is usually denoted by ∞ M A(i).
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