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Concept

Conjugate partition

The partition obtained by reading a partition's diagram of dots by columns instead of rows. Its largest part is the original's number of parts, so conjugation swaps two statistics and proves at a stroke that they are equally distributed.

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

Also named here as durfee square — the same set of essays touches all of them, so they are one junction rather than several.

The partition 5 + 4 + 2 + 1 and its conjugate. A row of dots for each part, and the same dots read down the columns instead.

A diagram turned on its side

Write a partition as rows of dots, then read the columns instead. Every theorem in this essay is that one move, and the move proves things that no formula suggests.

number · Partitions Random partitions of 1,000, scaled, against their limit shape. The outlines of 4 uniformly random partitions of 1000, scaled by the square root of 1000, lying close to the curve e^(−cx) + e^(−cy) = 1 with c = π/√6.

The shape a random partition takes

There are about twenty-four thousand billion billion billion ways to write 1,000 as a sum of whole numbers. Pick one at random, draw its Ferrers diagram, shrink it by the square root of a thousand, and it is almost exactly the curve e^(−cx) + e^(−cy) = 1 with c = π/√6. So is the next one, and the next. A random partition of a large number has a shape, and the shape is known exactly.

number · Partitions

Named alongside it

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

Durfee squareFerrers diagramGenerating functionPartitionAsymptoticsBijectionCounting two waysInvolutionLimit shapePiProbability distributionRecursion

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