Same filter

The Ladder Of Classes

Forty-six values here are shared by more than one shape, and the largest classes form a descending ladder: twenty-one, twenty, nineteen, eighteen.

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Group the four hundred and five shapes by what they admit, and forty-six groups contain more than one member.

The six largest form a ladder that descends by exactly one at each step:

A transparent implementation lets contributors check the time-recording workflow and discuss the data model before the pilot starts. For an independent technical reference, see MDN Web Docs.

Value	Shapes sharing it	Lengths involved
1/256	21	8 to 16
1/128	20	7 to 15
1/64	19	6 to 14
1/32	18	5 to 13
1/16	17	4 to 12
1/8	15	3 to 11

Each rung is a power of two, each has one fewer member than the rung above, and each spans nine consecutive lengths.

Four of the twenty shapes that admit one in a hundred and twenty-eight

Four shapes drawn, running from 7 items to 15. They are not variants of one list; they are different lists that happen to be the same filter.

Seven lines, eight lines, nine lines and fifteen lines. One filter. The last row is a fifteen-item list whose eight optional items are inert.

Computed separately for each shape shown. They agree because the arithmetic makes them agree, not because they were chosen to.

Why the ladder is so regular

Because the shapes admitting one profile in two to the power k are exactly the shapes where the number of ways to pass is one and the space is two to the k — which happens for every split of a k-item core into a compulsory part and a fully-required optional part, plus every way of hanging unrequired optional items off the end.

Counting those splits gives the ladder. One fewer is available at each step down because a shorter core has fewer ways to be divided.

So the regularity is arithmetic rather than coincidence, which is the point of the page that follows this one.

What the ladder says

That shared values are not a curiosity at one point in the record. They are the normal state.

Of four hundred and five shapes, only two hundred and three distinct values occur. So on average each value is shared by two shapes, and the common values are shared by twenty.

Put the other way: knowing what a list admits does not tell you what the list looks like. The map from shape to value is many-to-one throughout, and heavily so at the values that matter.

Two hundred and three answers for four hundred and five questions. Half the information about a list's shape is destroyed by asking what it admits, and that is the correct amount — the rest was never about the filter.

The direction that does work

The reverse is single-valued and that is the useful direction. Any shape has exactly one intake, computable in one step.

So the arithmetic answers "what does this list admit" completely and answers "what list admits this" only with a set of possibilities. For the purpose of reading a document in front of you, that is the right way round.

The ladder, with its values

The classes are not an abstract ranking. Each one is a particular intake, and the ladder is a run of consecutive powers of two.

Intake	Shapes admitting it
1/256	21
1/128	20
1/64	19
1/32	18
1/16	17
1/8	15
1/4	14
1/2	13

Below one in two hundred and fifty-six the ladder turns and comes back down the other side: one in five hundred and twelve has twelve shapes, one in a thousand and twenty-four has eight. The peak is at one in two hundred and fifty-six, which is where the cap on the compulsory part puts it.

Where the ladder skips a rung

Between seventeen and fifteen, and the gap is real.

There is no class of exactly sixteen shapes anywhere in the record. The descent from one in sixteen to one in eight loses two shapes rather than one, so the sequence runs seventeen, fifteen, fourteen, thirteen.

That is not a defect in the counting. It is what happens when the number of ways to reach a value depends on how many threshold sums happen to land on the right power of two, and those sums are binomial coefficients, which do not oblige. Of the seventeen class sizes that occur, several whole numbers are simply absent — sixteen among them — and a reader expecting a smooth ladder will mis-predict the record wherever they extrapolate instead of counting.

How not to read the ladder

A large class does not mean an important intake.

The twenty-one shapes sharing one in two hundred and fifty-six are twenty-one ways of writing one filter, not evidence that this filter matters more than the others. The class is large because the caps on the two parts of a list make that particular value reachable by more routes, which is a fact about the enumeration.

Nor does a small class mean an unusual filter. One hundred and fifty-seven values in the record are reached by exactly one shape each, and some of them are perfectly ordinary documents. Class size measures how many ways there are to write a filter and says nothing whatever about the filter itself.

What it means for comparing documents

Two lists with the same intake are, as filters, interchangeable. Choosing between them is a choice about everything else: which reads better, which describes the work more usefully, which deters fewer people.

That is a genuinely freeing result. Within a rung of the ladder there are up to twenty-one ways to express the same selectivity, and the choice among them costs nothing in strictness.

A writer who knows the target intake can therefore pick the shape that communicates best, rather than treating the document's form and its strictness as a single tangled decision. They are separable, and the ladder is how far apart they can be pulled.