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 $𝒜$ be a family of subsets of the set $A$ and let $x\in A$ be a distinguished element of set $A$. Then suppose there is a predicate $P(X,x)$ that relates a subset $X\subseteq A$ to $x$. Denote $𝒜(x)$ to be the set of subsets $X$ from $𝒜$ for which $P(X,x)$ is true and $𝒜-x$ to be the set of subsets $X$ from $𝒜$ for which $P(X,x)$ is false, Then $𝒜(x)$ and $𝒜-x$ are disjoint sets, so by the method of summation, the cardinalities are additive

$|𝒜|=|𝒜(x)|+|𝒜-x|$

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 $(\genfrac{}{}{0ex}{}{n}{k})$ 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:

$(\genfrac{}{}{0ex}{}{n}{k-1})+(\genfrac{}{}{0ex}{}{n}{k})=(\genfrac{}{}{0ex}{}{n+1}{k}).$

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 $(\genfrac{}{}{0ex}{}{n}{k-1})$. 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 $(\genfrac{}{}{0ex}{}{n}{k})$.

• 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:

$\begin{array}{c}abc\\ a/bc\\ b/ac\\ c/ab\\ a/b/c\end{array}$

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

$B_n+C_n=B_{n+1}.$

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