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Concept

Generating function

A series whose coefficients are the terms of a counting sequence. Multiplying two of them convolves the coefficients, so combining independent choices becomes multiplication and a recursion becomes an equation that can be solved.

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 equation the objects satisfy. A diagram of the decomposition C = 1 + xC², with a table of the first several coefficients computed two ways: by the convolution the equation prescribes, and from the closed form.

The equation a sequence satisfies

Write the whole sequence as the coefficients of one series, and the recursion becomes an equation with a square in it. Solving the equation by the ordinary quadratic formula produces the closed form, the growth rate and the correction term, none of which the recursion offers.

discrete · Catalan numbers

Named alongside it

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

PartitionCounting two waysRecursionBijectionAlgebraic identityAsymptoticsBinomial coefficientCatalan numbersConvolutionFormal power seriesInvolutionAnalytic continuation

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