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Concept

Convolution

The sequence whose n-th term sums the products of terms of two sequences whose indices add to n. It is what a product of generating functions computes, which is why a recurrence involving a split becomes an algebraic equation.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

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 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 35 routes across a 4 by 3 grid. A grid with each cell holding the number of monotone routes reaching it. The far corner holds 35, which is the binomial coefficient of 7 choose 3.

Every entry counts the routes to it

Turn Pascal's triangle forty-five degrees and it becomes a grid of street corners, with each entry counting the ways of walking there. Identities between the entries then become statements about routes, and the statements are proved by cutting the routes in one place.

discrete · Pascals triangle The coefficients of (1 + x)¹² sorted by remainder mod 3. The binomial coefficients of the 12th power coloured by the remainder of their index on division by 3, beside the 3 points one plus a root of unity, whose powers averaged pick out each colour's total.

Every third coefficient

Add every third number in the twelfth row of Pascal's triangle and the answer is 1366 — a third of 4096, rounded up. Which way the rounding goes is decided by two arrows of length one in the complex plane, and the same average over the roots of unity counts dice totals, subsets and necklaces.

algebra · Roots of unity The 6 ways to deal 4 labels between pieces of size 2 and 2. Every way of splitting 4 labels between a piece of size 2 and a piece of size 2, listed as two rows of boxes each. The count is the binomial coefficient that distinguishes a labelled product from an ordinary one.

The product that deals the labels

Multiplying two counting series pairs one choice with another. When the things being counted carry labels, the labels have to be dealt out as well, and the only series that survive the extra bookkeeping are the ones divided by n factorial.

discrete · Generating functions Products of widening sinc waves, all with area π/2 but the last. Graphs of the products of sinc(x/(2k+1)) for k = 0 to n, n = 0, 1, 3, 7, on 0 ≤ x ≤ 24.

Seven times half of π, and then not

The area under sin x / x, out to infinity, is π/2. Multiply in sin(x/3)/(x/3) and the area is still π/2; multiply in the waves for 5, 7, 9, 11 and 13 and it is still π/2, exactly. Add the wave for 15 and the area falls short — by about 2.3 × 10⁻¹¹. Nothing about the curves changes at the eighth factor. What changes is that 1/3 + 1/5 + … + 1/15 has just passed 1.

analysis · Circular functions

Named alongside it

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

Binomial coefficientGenerating functionCounting two waysBijectionCatalan numbersFormal power seriesRecursionAlgebraic identityAsymptoticsCombinatorial proofCyclic groupDerangement

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