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Concept

Formal power series

An infinite sequence of coefficients written with powers of a variable as position markers, added and multiplied by the usual rules. Nothing is ever substituted for the variable, so convergence never arises, and each coefficient of a product is settled by finitely many terms.

Named by 3 essays across 2 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 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 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

Named alongside it

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

Counting two waysGenerating functionBijectionBinomial coefficientConvolutionPartitionAlgebraic identityCatalan numbersCoefficientDerangementGeometric seriesPermutation

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