Bijective proof

In combinatorics, bijective proof is a proof technique that finds a bijective function (that is, a one-to-one and onto function) $f : A \to B$ between two finite sets $A$ and $B$ , or a size-preserving bijective function between two combinatorial classes, thus proving that they have the same number of elements, $| A | = | B |$ . One place the technique is useful is where we wish to know the size of $A$ , but can find no direct way of counting its elements. By establishing a bijection from $A$ to some $B$ solves the problem if $B$ is more easily countable. Another useful feature of the technique is that the nature of the bijection itself often provides powerful insights into each or both of the sets.

Basic examples

Proving the symmetry of the binomial coefficients

The symmetry of the binomial coefficients states that

{n \choose k}={n \choose n-k}.

This means that there are exactly as many combinations of $k$ things in a set of size $n$ as there are combinations of $n - k$ things in a set of size $n$ .

A bijective proof

The key idea of the proof may be understood from a simple example: selecting out of a group of $n$ children which $k$ to reward with ice cream cones has exactly the same effect as choosing instead the $n - k$ children to be denied them.

More abstractly and generally,^[1] the two quantities asserted to be equal count the subsets of size $k$ and $n - k$ , respectively, of any $n$ -element set $S$ . Let $A$ be the set of all $k$ -element subsets of $S$ , the set $A$ has size ${\tbinom {n}{k}}.$ Let $B$ be the set of all $n-k$ subsets of $S$ , the set B has size ${\tbinom {n}{n-k}}$ . There is a simple bijection between the two sets $A$ and $B$ : it associates every $k$ -element subset (that is, a member of $A$ ) with its complement, which contains precisely the remaining $n - k$ elements of $S$ , and hence is a member of $B$ . More formally, this can be written using functional notation as, $f : A \to B$ defined by $f (X) = X c$ for $X$ any $k$ -element subset of $S$ and the complement taken in $S$ . To show that f is a bijection, first assume that $f (X 1) = f (X 2)$ , that is to say, $X 1 c = X 2 c$ . Take the complements of each side (in $S$ ), using the fact that the complement of a complement of a set is the original set, to obtain $X 1 = X 2$ . This shows that $f$ is one-to-one. Now take any $n-k$ -element subset of $S$ in $B$ , say $Y$ . Its complement in $S$ , $Y c$ , is a $k$ -element subset, and so, an element of $A$ . Since $f (Y c) = (Y c) c = Y$ , $f$ is also onto and thus a bijection. The result now follows since the existence of a bijection between these finite sets shows that they have the same size, that is, ${\tbinom {n}{k}}={\tbinom {n}{n-k}}$ .

Other examples

Problems that admit bijective proofs are not limited to binomial coefficient identities. As the complexity of the problem increases, a bijective proof can become very sophisticated. This technique is particularly useful in areas of discrete mathematics such as combinatorics, graph theory, and number theory.

The most classical examples of bijective proofs in combinatorics include:

Prüfer sequence, giving a proof of Cayley's formula for the number of labeled trees.
Robinson-Schensted algorithm, giving a proof of Burnside's formula for the symmetric group.
Conjugation of Young diagrams, giving a proof of a classical result on the number of certain integer partitions.
Bijective proofs of the pentagonal number theorem.
Bijective proofs of the formula for the Catalan numbers.

References

^ Mazur, David R. (2010), Combinatorics / A Guided Tour, The Mathematical Association of America, p. 28, ISBN 978-0-88385-762-5

External links

"Division by three" – by Doyle and Conway.
"A direct bijective proof of the hook-length formula" – by Novelli, Pak and Stoyanovsky.
"Bijective census and random generation of Eulerian planar maps with prescribed vertex degrees" – by Gilles Schaeffer.
"Kathy O'Hara's Constructive Proof of the Unimodality of the Gaussian Polynomials" – by Doron Zeilberger.
"Partition Bijections, a Survey" – by Igor Pak.
Garsia-Milne Involution Principle – from MathWorld.

[1] Mazur, David R. (2010), Combinatorics / A Guided Tour, The Mathematical Association of America, p. 28, ISBN 978-0-88385-762-5

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