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Concept

Asymptotics

The behaviour of a quantity in the limit, stated as a ratio: two expressions are asymptotic when their quotient tends to one. That says nothing about their difference, which can grow without bound while the ratio settles, so an asymptotic formula needs an error term before it can be used at a particular size.

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

How many Latin squares there are, orders 1 to 8. The number of Latin squares of each small order, the ones up to six counted by exhaustive search and the larger ones quoted, on a logarithmic scale.

Nine thousand four hundred and eight

There are four Latin squares of order four once the first row and column are fixed, fifty-six of order five, and nine thousand four hundred and eight of order six. The exact answer is known for eleven orders and for no more — and yet a half-finished square can always be finished.

computation · latin squares p(n) to 60, against the Hardy–Ramanujan estimate. The number of partitions of each number up to sixty on a logarithmic scale, with the asymptotic estimate drawn over it and the ratio of the two tabulated.

The size of a number with no formula

There is no closed expression for the number of partitions of n. There is an expression for how large it is — with a square root in the exponent and a π in front — and it is accurate enough that rounding a few terms of its refinement gives the exact count.

number · partitions

Named alongside it

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

CountingAnalytic continuationApproximationBijectionConvergenceEstimateExhaustive searchFactorialGenerating functionGrowth rateHalls conditionLatin rectangle

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