Basic principles and techniques of enumerative combinatorics include:
The principles listed above are disarmingly simple and seemingly obvious. Nonetheless, when used properly they are powerful tools for producing bijective proofs of combinatorial identities. On the other hand, while generating functions can frequently be used to give quick proofs of identities, it is sometimes difficult to extract combinatorial proofs from such proofs.
The most fundamental principle of enumerative combinatorics and the basis of all counting is the addition principle. It says that if is a finite set, then
More generally, if , then
By induction on , we also get
As an application of the addition principle, we prove the multiplication principle.
Multiplication principle. If and are finite sets, then
The involution principle says that if is an involution, then for any , . The following example illustrates the involution principle.
In other words, is the symmetric difference of and — if is an element of , we remove it, and if is not an element of , we insert it. The function is an involution, hence a bijection. Now notice that and . In other words,
Since is a bijection, this implies that
By induction on we obtain
Now , so . Thus , implying that
which is what we wanted to show. ∎
As we have seen above, it is possible to use the addition principle to count the number of elements in the disjoint union of finite sets. But what if we want to count the number of elements in a non-disjoint union of finite sets? The inclusion-exclusion principle gives us a way to do this. A common way of stating the inclusion-exclusion principle is as the following proposition.
Inclusion-exclusion principle. Let be finite sets. Then
The formula in this proposition uses negative numbers. But the core of the inclusion-exclusion principle is a statement about natural numbers.
By the addition principle,
Now observe that . So by a second application of the addition principle,
Hence . ∎
|Date of creation||2013-03-22 16:45:58|
|Last modified on||2013-03-22 16:45:58|
|Last modified by||mps (409)|