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Concept

Reflection principle

The correspondence folding a path about the first moment it touches a level, which turns a question about a path's history into one about its endpoint. It counts paths that touch a barrier by counting all paths from a mirrored start, and it degrades as soon as there are two barriers.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

A path folded about the first time it touches. A walk from 2 to 4 that touches the axis, with the part before its first touch reflected. The reflection is a path from the mirrored start to the same endpoint, and the correspondence is exact.

The path folded at its first touch

Counting the walks that touch a line looks like a question about a walk's whole history. Fold each one where it first touches, and it becomes a question about where walks end up — which is a binomial coefficient, and is already known.

probability · Random walk Time spent on one side of the axis. The exact distribution of the number of steps a 40-step fair walk spends above the axis. It is U-shaped: the extremes are the likeliest outcomes and an even split is the rarest.

Half the time is the rarest answer

In a fair game of many rounds, the fraction of the time one side is ahead is not usually near a half. It is usually near nought or one, and an even split is the single least likely outcome there is.

probability · Random walk A bad path, and the path it reflects to. Two grids, 6 by 5. On the left a monotone path that dips below the diagonal, with its first offending step marked; on the right the same path with everything after that step reflected, which ends one square right and one square below the corner.

Counting the paths that go wrong

The number of good paths across a grid has no obvious formula. The number of bad ones does, because every bad path can be reflected into a path to a different corner, and that reflection is a perfect matching between two sets nobody chose to relate.

discrete · Catalan numbers

Named alongside it

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

Random walkBallot problemBijectionBinomial coefficientFirst returnArcsine lawCatalan numbersCentral limitCounting-two waysDistributionFairnessLattice paths

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