Cayleys formula
Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.
Sixteen trees on four points
How many ways are there to connect n labelled points into a single tree? The answer is n to the power n minus two, which is a strange enough formula to demand an explanation — and the explanation is a code that turns every tree into a short list of numbers, and every short list of numbers back into a tree.
Cars that park, and trees that grow
Three cars arrive at a one-way street with three spaces; each has a favourite space, drives to it, and takes the first free one from there on. Of the 27 lists of favourites, exactly 16 let every car park — the same 16 as the labelled trees on four points. The reason is a circular street with one extra space, on which every list parks and exactly one rotation of it leaves the extra space empty.
The cheapest tree costs ζ(3)
Join n towns by every possible road, give each road a cost drawn at random between nought and one, and find the cheapest network of n − 1 roads that connects them all. However many towns there are, the bill tends to 1 + 1/8 + 1/27 + 1/64 + … = 1.2020569. The proof watches the cheap roads arrive one at a time and turns the cost into the area under a curve that counts the pieces of a growing random graph.
How many shapes a tree can take
There are 262,144 labelled trees on eight points and only twenty-three shapes. Labels make counting easy — Cayley's nⁿ⁻² — and shapes make it hard, because a symmetric shape absorbs labellings. The count of shapes comes instead from a recurrence for trees with a root and a theorem of Richard Otter's for taking the root away: in every tree there is exactly one more class of points than of edges. Run to two hundred points, the counts grow like 0.535 × 2.9558ⁿ ÷ n^(5/2), and the same arithmetic counts the alkanes.
The forests a graph can hold
Count the sets of edges in a graph that contain no cycle, sorted by size, and the counts rise and fall in a single peak — more sharply peaked, John Mason conjectured in 1972, than the binomial coefficients with the same number of edges. On every graph with up to six points the strongest of his three inequalities holds, with equality exactly where cycles are too long to matter; on random graphs with nine points it holds with a margin of about one per cent. It was proved for every graph, and every matroid, in 2018.
Named alongside it
The objects these essays reach for when they reach for this one.
Spanning treeBijectionCounting argumentExhaustive searchGenerating functionLabelled treeAsymptoticsAutomorphismBurnside lemmaCatalan numbersCentral limit theoremChromatic polynomial