In the mathematical field of enumerative combinatorics, identities are sometimes established by arguments that rely on singling out one "distinguished element" of a set.

Definition

be a family of subsets. Then suppose there is a predicate are disjoint sets, so by the method of summation, the cardinalities are additive

Thus the distinguished element allows for a decomposition according to a predicate that is a simple form of a divide and conquer algorithm. In combinatorics, this allows for the construction of recurrence relations. Examples are in the next section.

Examples

• The binomial coefficient is the number of size-k subsets of a size-n set. A basic identity—one of whose consequences is that the binomial coefficients are precisely the numbers appearing in Pascal's triangle—states that:

• Proof: In a size-(n+1) set, choose one distinguished element. The set of all size-k subsets contains: (1) all size-k subsets that do contain the distinguished element, and (2) all size-k subsets that do not contain the distinguished element. If a size-k subset of a size-(n+1) set does contain the distinguished element, then its other k1 elements are chosen from among the other n elements of our size-(n+. If a size-k subset does not contain the distinguished element, then all of its k members are chosen from among the other n.

• The number of subsets of any size-n set is 2n.

• Proof: We use mathematical induction. The basis for induction is the truth of this proposition in case n=0. The empty set has 0 members and 1 subset, and 20=1. The induction hypothesis is the proposition in case n; we use it to prove case n+1. In a size-(n+1) set, choose a distinguished element. Each subset either contains the distinguished element or does not. If a subset contains the distinguished element, then its remaining elements are chosen from among the other n elements. By the induction hypothesis, the number of ways to do that is 2n. If a subset does not contain the distinguished element, then it is a subset of the set of all non-distinguished elements. By the induction hypothesis, the number of such subsets is 2n. Finally, the whole list of subsets of our size-(n+1) set contains 2n+2n=2n+1 elements.

• Let Bn be the nth Bell number, i.e., the number of partitions of a set of n members. Let Cn be the total number of "parts" (or "blocks", as combinatorialists often call them) among all partitions of that set. For example, the partitions of the size-3 set {a,b,c} may be written thus:

• We see 5 partitions, containing 10 blocks, so B3=5 and C3=10. An identity states:

  Proof: In a size-(n+1) set, choose a distinguished element. In each partition of our size-(n+1) set, either the distinguished element is a "singleton", i.e., the set containing only the distinguished element is one of the blocks, or the distinguished element belongs to a larger block. If the distinguished element is a singleton, then deletion of the distinguished element leaves a partition of the set containing the n non-distinguished elements. There are Bn ways to do that. If the distinguished element belongs to a larger block, then its deletion leaves a block in a partition of the set containing the n non-distinguished elements. There are Cn such blocks.

See also

• Combinatorial principles

• Combinatorial proof

References