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Counting the same thing twice — page 11

One collection, counted by two different methods, and an identity that falls out because both answers have to agree. The proof is the pair of counts.

A function on nine points, and the tree with two marks it becomes. Left: a function on 9 points as arrows, with cycles through 2, 5, 7 and trees hanging into them. Right: the tree Joyal's rule makes from it, a path from head 7 to tail 2 with the same trees hanging below.

Discrete

Every function is a tree with two marks

There are n to the n functions from n points to themselves, and n to the n − 2 trees on those points. André Joyal noticed in 1981 that the missing factor of n² is a choice of two points — a head and a tail — and that a function, read the right way, simply is a tree with a head and a tail. The reading turns Cayley's formula into one line and hands over a fact about random trees from the birthday problem.

The 11 trees on 7 points, and how many labellings each has. All trees on 7 points with their numbers of symmetries and of distinct labellings; the labellings add to 16807.

Discrete

Almost every tree can be turned over

Cayley's n^(n−2) counts trees with labels on their points. Take the labels off and the count has no formula, because a symmetric shape absorbs labellings: the star on seven points can be labelled only seven ways, the one asymmetric shape 5,040. The bookkeeping that reconciles the two counts says something unexpected — almost every tree, labelled or not, can be turned over onto itself, where almost every graph cannot.

Two ways to cut a hexagon, one total of radii. A cyclic hexagon triangulated two ways with the incircles drawn; the inradii sum to 1.1622 both times.

Geometry

A total hung in a temple

Put a polygon's corners on a circle, cut it into triangles, and add up the radii of the circles inscribed in the triangles. Cut it a different way and the total is the same — for all fourteen ways of cutting a hexagon, to every digit. The fact was painted on a wooden tablet in a Japanese temple around 1800, and the reason for it is a theorem about one triangle and the distances from its circumcentre to its sides.

Four lists, merged in pairs, find a cancelling quadruple. A merge tree for four lists of 256 random 24-bit strings: two merges on the low 8 bits, then one on all 24, ending in 1 solution(s).

Probability

Four lists find a collision sooner

Two lists of random 24-bit strings need about four thousand entries each before some string appears in both — the birthday bound. Ask instead for one string from each of four lists whose exclusive-or is zero, and lists of a few hundred suffice, because merging the lists in pairs manufactures the near-misses a direct search would have to find by luck. David Wagner's algorithm turns the square root into a cube root, and more lists into smaller roots.

One shared birthday, as a measurement of the year. Histogram over 20000 rooms of the estimate T(T − 1)/2 from the first shared birthday, averaging 361 against 365, spread widely.

Probability

Counting a population by its repeats

The birthday problem runs forwards from a known year to the chance of a match. Run it backwards and the matches measure the year: count how many pairs among the draws came out the same, and the number of kinds is about the number of pairs divided by that. It costs the square root of the population rather than a census of it — and when the kinds are unevenly common, what the repeats measure is not how many kinds there are but how many they behave like.

A loop that cannot be undone, and the surface it bounds. The loop aba⁻¹b⁻¹ on the figure eight, which cannot be pulled tight, beside a square whose boundary reads the same word, showing that the loop bounds a surface.

Topology

What homology forgets about a loop

Let the letters of a loop commute and the loop group of a space becomes its first homology group: a loop now records only how often it went round each hole. What is thrown away is exactly the loops that bound a surface. On the figure eight that is nearly everything — of the loops of sixty letters that homology calls nought, about one in 6,700 is a loop that actually shrinks.

Pascal's triangle mod 2, and the room for the 2-dimensional projective space. Pascal's triangle with odd entries filled, rows 0 to 15, with row 3 marked as the tangent ledger 1 + a + a² and row 1 as the normal ledger 1 + a.

Topology

The room a projective space needs, read off Pascal's triangle

The projective plane cannot sit in three-dimensional space without crossing itself, and the reason can be written as arithmetic: a polynomial that records how a shape twists, which a room must cancel. For the n-dimensional projective space that polynomial is a row of Pascal's triangle read mod 2, its inverse is another row, and the inverse's last term says how many extra dimensions the room must have — exactly enough, at every power of two.

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