Concept

Sign — where it appears

The single bit a permutation carries: whether it is built from an even or an odd number of swaps. It is the only label on permutations that multiplies correctly into a commutative group, and it is what makes a determinant possible.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

The permutation (1 3 4 2) drawn as 4 strings, crossing 3 times. A permutation drawn as strings running from a row of numbered pegs to another, with every place two strings cross marked, and the crossing count checked against the number of pairs that are out of order.

The crossings that will not come out even

Draw a rearrangement as strings from one row of pegs to another and count where they cross. The count depends on how the strings are drawn; whether it is odd or even does not, and that single bit is what makes determinants exist and a sliding puzzle unsolvable.

algebra · Permutation parity

The character table of the permutations of 4 places, computed from traces. A table with one row per irreducible representation and one column per conjugacy class, giving the trace of the matrix each representation assigns, with the dimension column marked.

When the label may be a matrix

A permutation carries exactly one bit into any commutative target, and commutativity is the restriction doing all the work. Drop it — let the label be a matrix — and what survives is a short finite table, computed here from traces and checked for orthogonality over every pair of rows.

algebra · Permutation parity

Counting the real roots of x⁵ − 5x³ + x² + 3x − 1 with Euclid's algorithm. Above, the graph of a polynomial over an interval with its real roots marked; below, a step function counting sign changes in the polynomial's Sturm chain, which drops by one at each root; beside them the chain of polynomials listed.

The remainders that count the roots

Run Euclid's algorithm on a polynomial and its derivative, flipping the sign of each remainder, and write down the signs of the whole chain at any point. The number of sign changes drops by exactly one each time the point passes a real root — so the roots in any interval can be counted, exactly, without finding a single one.

geometry · Euclidean algorithm

The proof of Descartes' rule, one term at a time. Four stacked graphs: a four-term polynomial with three positive roots, and the three polynomials obtained by dividing out a power of x and differentiating, each with one fewer term, one fewer sign change and one fewer positive root.

What the signs allow

Descartes' rule reads the signs of a polynomial's coefficients and bounds its positive roots by how often they change — a bound set by the number of terms, not the degree, and proved by nothing more than Rolle's theorem. Each count the rule allows can be had. But not every pair of counts, positive and negative, that the rule allows together can be had together: the first combination that never occurs is at degree four, and the list of impossible ones is still being worked out.

geometry · Euclidean algorithm

Named alongside it

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

Derivative Exact arithmetic Parity Permutation Polynomial Roots Bijection Character table Conjugacy class Counting argument Descartes rule of signs Determinant

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