Permutation
Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.
Eight ways to leave a square alone
A square can be picked up and put back so that nothing looks different. There are exactly eight ways to do it, and the number is not asserted here — it is what a search through all twenty-four relabellings of the corners comes back with.
Colourings nobody can tell apart
Sixteen ways to colour four corners in two colours, and only six of them are genuinely different. The count can be got by pooling the sixteen — or by never forming a single class and instead averaging how many colourings each motion leaves untouched.
Nobody gets their own hat
Hand back a pile of hats at random and ask for the chance that not one person gets their own. The answer barely moves as the crowd grows — it is a third and a bit at four people, and a third and a bit at four thousand.
Named alongside it
The objects these essays reach for when they reach for this one.
Counting argumentCounting two waysCyclic groupDihedral groupGroup actionLagrange theoremSymmetryAlternating seriesApproximationClosureComplementary countingDerangement