Concept

Algebraic identity

An equation that holds for every value of the letters in it rather than for particular ones, so that both sides are two descriptions of the same object. Proving one usually means transforming one side into the other, and a counting identity is a claim that two ways of counting the same collection agree.

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

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

Named alongside it

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

Bijection Counting two ways Generating function Partition Recursion Cancellation Coefficient Formal power series Geometric series Involution Parity Pentagonal number

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