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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 4 essays across 2 fields — 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 A run down a diagonal, and the entry it adds to. 9 rows of Pascal's triangle with 5 entries shaded and the entry they add to marked. The claim is checked by adding the shaded entries: 1 + 3 + 6 + 10 + 15 = 35.

The run that lands one place along

Add up a run of entries down one of Pascal's diagonals and the total is another entry of the triangle — one row further down and one place along. The same triangle holds four more sums of that kind, and each is a different question answered by the same additive rule.

discrete · Pascals triangle 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

Named alongside it

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

Counting two waysRecursionBijectionBinomial coefficientCombinatorial proofGenerating functionPartitionCancellationCoefficientConvolutionFibonacciFormal power series

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