I’m currently at the training camp in Cambridge for this year’s UK IMO squad. This afternoon I gave a talk to some of the less experienced students about combinatorics. My aim was to cover as many useful tricks for calculating the sizes of combinatorial sets as I could in an hour and a half. We started by discussing binomial coefficients, which pleasingly turned out to be revision for the majority. But my next goal was to demonstrate that we are much more interested in the fact that we can calculate these if we want than in the actual expression for their values.
Put another way, my argument was that the interpretation of as the number of ways to choose m objects from a collection of n, or the number of up-and-right paths from (0,0) to (m,n) is more useful than the fact that . The opening gambit was to prove the fundamental result underlying the famous construction of Pascal’s triangle that
This is not a hard result to prove by manipulating factorials, but it is a very easy result to prove in the path-counting setting, for example.
So it turned out that the goal of my session, as further supported by some unsubtly motivated problems from the collection, was to convince the students to use bijections as much as possible. That is, if you have to count something awkward, show that counting the awkward thing is equivalent to counting something more manageable, then count that instead. For many simpler questions, this equivalence is often drawn implicitly using words (“each of the n objects can be in any subset of the collection of bags so we multiply…” etc), but it is always worth having in mind the formal bijective approach. Apart from anything else, asking the question “is this bijection so obvious I don’t need to prove it” is often a good starting-point for assessing whether the argument is in fact correct!
Anyway, I really wanted to show my favouriite bijection argument, but there wasn’t time, and I didn’t want to spoil other lecturers’ thunder by defining a graph and a tree and so forth. The exploration process encoding of trees is a strong contender, but today I want to define quickly the Prufer coding for trees, and use it to prove a famous result I’ve been using a lot recently, Cayley’s formula for the number of spanning trees on the complete graph with n vertices, .
We are going to count rooted trees instead. Since we can choose any vertex to be the root, there are rooted trees on n vertices. The description of the Prufer code is relatively simple. Take a rooted tree with vertices labelled by [n]. A leaf is a vertex with degree 1, other than the root. Find the leaf with the largest label. Write down the label of the single vertex to which this leaf is connected, then delete the leaf. Now repeat the procedure, writing down the label of the vertex connected to the leaf now with the largest label, until there are only two vertices remaining, when you delete the non-root vertex, and write down the label of the root. We get a string of (n-1) labels. We want to show that this mapping is a bijection from the set of rooted trees with vertices labelled by [n] to .
Let’s record informally how we would recover a tree from the Prufer code. First, observe that the label of any vertex which is not a leaf must appear in the code. Why? Well, the root label appears right at the end, if not earlier, and every vertex must be deleted. But a vertex cannot be deleted until it has degree one, so the neighbours further from the root (or ancestors) of the vertex must be removed first, and so by construction the label appears. So know what the root is, and what the leaves are straight away.
In fact we can say slightly more than this. The number of times the root label appears is the degree of the root, while the number of times any other label appears is the degree of the corresponding vertex minus one. Call this sequence the Prufer degrees.
So we construct the tree backwards from the leaves towards the root. We add edges one at a time, with the k-th edge joining the vertex with the k-th label to some other vertex. For k=1, this other vertex is the leaf with maximum label. In general, let be the graph formed after the addition of k-1 edges, so is empty, and is the full tree. Define to be the set of vertices such that their degree in is exactly one less than their Prufer degree. Note that is therefore the set of leaves suggested by the Prufer code. So we form by adding an edge between the vertex with label appearing at position k+1 in the Prufer sequence and the vertex of with maximum label.
Proving that this is indeed the inverse is a bit fiddly, more because of notation than any actual mathematics. You probably want to show injectivity by an extremal argument, taking the closest vertex to the root that is different in two trees with the same Prufer code. I hope it isn’t a complete cop out to swerve around presenting this in full technical detail, as I feel I’ve achieved by main goal of explaining why bijection arguments can reduce a counting problem that was genuinely challenging to an exercise in choosing sensible notation for proving a fairly natural bijection.