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Concept

Recurrence relation

A rule giving each term of a sequence from the terms before it, with enough starting values to begin. Solving one turns a step-by-step rule into a formula, and the solutions of a linear one are powers of the roots of its characteristic equation.

Named by 5 essays across 4 fields — each of them below, with the objects they name alongside it.

A walk with a barrier at each end. Three games played to absorption on a table of 12, beside the chance of ruin from each starting stake — a straight line, because the walk is fair.

Two barriers and a fair game

A fair walk between two absorbing barriers is ruined with a probability that is a straight line in the starting stake, and lasts for a number of steps that is the product of what each side can lose. Both facts come from the same two-line recurrence, and both are bad news for the smaller player.

probability · Random walk Counting the colourings, by deleting and contracting. A graph beside the two smaller graphs its edge deletion and contraction produce, with the number of proper colourings of each at every number of colours up to 5.

Counting the colourings

Asking whether a graph can be coloured with four colours gives a yes or a no. Asking how many ways there are gives a polynomial — and the polynomial answers the first question, and several others nobody asked.

discrete · Graph colouring 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 Apéry's proof for ζ(3), as three numbers. log₁₀ of 2·lcm(1, …, n)³, of |a(n)ζ(3) − b(n)| and of their product for n up to 60: slopes 1.262, -1.546, -0.284 per step.

The race that makes ζ(3) irrational

Roger Apéry's 1978 proof that the sum of the reciprocal cubes is not a fraction comes down to a race between two numbers. A whole-number multiplier grows by a factor of ten every 0.79 steps; the gap it multiplies shrinks by a factor of ten every 0.65. The gap wins, by a margin of about seventeen per cent, and that margin is the whole proof.

number · Irrationality The 16 zigzag arrangements of 5 numbers. All alternating permutations of 1..5: 13254, 14253, 14352, 15243, 15342, 23154, 24153, 24351, 25143, 25341, 34152, 34251, 35142, 35241, 45132, 45231.

The tangent counts zigzags

Arrange the numbers 1 to n so that they go up, down, up, down. There are 1, 1, 2, 5, 16, 61, 272 such arrangements, and those are, multiplied by factorials, the Taylor coefficients of tan x and sec x — the odd lengths from the tangent, the even from the secant. Désiré André proved it in 1879 by splitting a zigzag at its largest number, and the split turns out to be the derivative of the tangent written out term by term.

analysis · Circular functions

Named alongside it

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

Absorbing stateBinomial coefficientCatalan numbersChromatic numberChromatic polynomialConvolutionCountingCounting two waysDeletion contractionDifferential equationExpectationFormal power series

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