Concept

Cancellation

The disappearance of terms of opposite sign from a sum, leaving fewer than were written down. It is what makes a determinant computable by elimination and a signed count exact, and its absence is exactly why the same sum with the signs deleted is far harder to evaluate.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

$The fractions, put in a line. A grid whose rows are numerators and columns denominators, walked by antidiagonals, with the place each fraction takes in the list written in its cell and the repeats left blank.$

The arithmetic that loses subtraction

Adding one to an infinite collection changes nothing, and neither does doubling it, or squaring it. What that costs is the two operations that were doing the work — an equation between infinite sizes cannot be cancelled, and how many are left stops being a question.

logic · cardinality

A determinant counting the 16 spanning trees. A small graph, the minor of its Laplacian, and every one of its spanning trees drawn as thumbnails.

A determinant that counts trees

Write down a graph's Laplacian, strike out one row and its column, take the determinant. The answer is the number of spanning trees — and the minus signs in the determinant are what cancel every subset of edges that is not one.

algebra · determinant

A determinant of −5 and a permanent of 23 from the same six products. The six products of a three-by-three matrix listed once, added with signs to give the determinant and without signs to give the permanent, with a row operation applied to both.

The same sum without its minus signs

Delete the signs from the determinant's sum over permutations and what is left counts things directly rather than by cancellation. It is a better count and a far worse object — because the cancellation was what made the determinant computable.

algebra · determinant

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 argument Counting two ways Determinant Involution Matrix Absorption Algebraic identity Bounds Cardinal arithmetic Countability Generating function