Method of distinguished element

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

Let [math]\displaystyle{ \mathcal{A} }[/math] be a family of subsets of the set [math]\displaystyle{ A }[/math] and let [math]\displaystyle{ x \in A }[/math] be a distinguished element of set [math]\displaystyle{ A }[/math]. Then suppose there is a predicate [math]\displaystyle{ P(X,x) }[/math] that relates a subset [math]\displaystyle{ X\subseteq A }[/math] to [math]\displaystyle{ x }[/math]. Denote [math]\displaystyle{ \mathcal{A}(x) }[/math] to be the set of subsets [math]\displaystyle{ X }[/math] from [math]\displaystyle{ \mathcal{A} }[/math] for which [math]\displaystyle{ P(X,x) }[/math] is true and [math]\displaystyle{ \mathcal{A}-x }[/math] to be the set of subsets [math]\displaystyle{ X }[/math] from [math]\displaystyle{ \mathcal{A} }[/math] for which [math]\displaystyle{ P(X,x) }[/math] is false, Then [math]\displaystyle{ \mathcal{A}(x) }[/math] and [math]\displaystyle{ \mathcal{A}-x }[/math] are disjoint sets, so by the method of summation, the cardinalities are additive^[1]

[math]\displaystyle{ |\mathcal{A}| = |\mathcal{A}(x)| + |\mathcal{A}-x| }[/math]

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 [math]\displaystyle{ {n \choose k} }[/math] 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:

[math]\displaystyle{ {n \choose k-1}+{n \choose k}={n+1 \choose k}. }[/math]

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 k − 1 elements are chosen from among the other n elements of our size-(n + 1) set. The number of ways to choose those is therefore [math]\displaystyle{ {n \choose k-1} }[/math]. If a size-k subset does not contain the distinguished element, then all of its k members are chosen from among the other n "non-distinguished" elements. The number of ways to choose those is therefore [math]\displaystyle{ {n \choose k} }[/math].

• 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 B_n be the nth Bell number, i.e., the number of partitions of a set of n members. Let C_n 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:

[math]\displaystyle{ \begin{matrix}abc \\ a/bc \\ b/ac \\ c/ab \\ a/b/c \end{matrix} }[/math]

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

[math]\displaystyle{ B_n+C_n=B_{n+1}. }[/math]

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 B_n 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 C_n such blocks.

See also

• Combinatorial principles
• Combinatorial proof

References

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