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What Is Enumerative Combinatorics?
1.1 How to Count
The basic problem of enumerative combinatorics is that of counting the number of elements of a finite set.- P1
1.1.4 Example. There are actually formulas in the literature ("nameless here forevermore") for certain counting functions f(n) whose evaluation requires listingall (or almost all) of the f(n) objects being counted! Such a "formula" is completely worthless.- P2
Example 1.1.5 provides a simple illustration of the general principle that, informally speaking, if we have an identity involving power series that is valid when thepower series are regarded as functions (so that the variables are sufficiently smallcomplex numbers), then this identity continues to remain valid when regarded asan identity among formal power series, provided the operations defined in the formulas are well defined for formal power series. It would be unnecessarily pedanticfor us to state a precise form of this principle here, since the reader should havelittle trouble justifying in any particular case the formal validity of our manipulations with power series.- P6
