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Concept

Partition

A way of writing a whole number as a sum of whole numbers with the order disregarded. The counts have no simple closed form and are handled instead through a product whose factors decide how many copies of each part are used.

Named by 7 essays across 2 fields — each of them below, with the objects they name alongside it.

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 A product of 3 polynomials, and what its coefficients count. The coefficients of a product of small polynomials, with the combinations of choices that reach one marked total written out beneath it.

A polynomial that counts

Hang a counting sequence on the powers of a variable and the two ways of combining choices — this and that, this or that — become multiplication and addition, so a recursion turns into an equation and the equation can be solved.

discrete · generating functions The partition product's coefficients to q¹². A row of series coefficients computed by expanding a product, beside the same numbers obtained another way.

Every partition, hidden in a product

Multiply out one factor for each part size and the coefficient of q to the n is the number of partitions of n. Nothing is being approximated: the product is a bookkeeping device that does the counting by multiplying.

number · partitions The product of (1 − qᵏ), and what survives at 12. The coefficients of the pentagonal product drawn as signed bars, with the partitions into distinct parts that Franklin's move leaves unpaired.

The terms that cancel almost everything

Multiply out the product of 1 − q, 1 − q², 1 − q³ and so on, and nearly every coefficient is zero. What survives is a single plus or minus one at 1, 2, 5, 7, 12, 15 — and the reason is a way of pairing partitions off so that each pair cancels.

number · partitions 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 Every fifth partition count divides, and the rank that says why. A row of partition counts with the ones in a congruence class marked, and a histogram of partitions sorted by rank.

Every fifth one divides

p(4) is 5, p(9) is 30, p(14) is 135, and every partition count at a number leaving four on division by five is divisible by five. Ramanujan read it off a table; the explanation is a way of splitting those partitions into five equal heaps.

number · partitions The cube cut into 6 symmetric chains. The subsets of a set of 4 partitioned into 6 chains by the bracket rule, each chain running from size k to size 4 − k and passing once through the middle layer.

The cube cut into chains

Write a subset as a string of brackets, match them the ordinary way, and the unmatched ones say which chain it is on. Six chains cover all sixteen subsets of a four-element set, and the bound and the example arrive together.

discrete · posets

Named alongside it

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

Generating functionBijectionCounting two waysRecursionAlgebraic identityBinomial coefficientFormal power seriesInvolutionAnalytic continuationAntichainApproximationAsymptotics

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