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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 2 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

Named alongside it

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

Counting two waysGenerating functionPartitionAlgebraic identityBijectionBinomial coefficientCatalan numbersCoefficientConvolutionGeometric seriesProductRecurrence relation

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