Concept

Labelled tree

A tree — a connected network with no cycles — whose vertices carry distinct names, so two trees are the same only if the same pairs of names are joined. There are n^(n−2) of them on n names, Cayley's formula, and the labels are what make that count finite and exact.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

All 16 trees on 4 labelled points. Every tree on 4 labelled points, drawn one by one. There are 16 of them, which is 4 to the power 2.

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.

discrete · Labelled trees

Orbit times stabiliser is 8, on every row. A table of 5 things the 8 symmetries of a 4-gon can move. Each row draws every position the thing can be carried to and lists the motions that leave it where it is; the two counts multiply to 8 on every row.

Twenty-four ways to set a cube down

Count the rotations of a cube from its corners and the answer is eight times three. Count from its edges and it is twelve times two; from its faces, six times four. Three different pictures give one number because each count is the same theorem — the places a thing can go, times the motions that leave it where it is — and the same theorem splits Cayley's sixteen trees into twelve and four and proves that a group of eight has a centre.

algebra · Symmetry groups

Named alongside it

The objects these essays reach for when they reach for this one.

Counting argument Automorphism Bijection Catalan numbers Cayleys formula Conjugacy class Coset Dihedral group Encoding Graph Group action Lagrange theorem

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