Length

The Narrowest List At Any Length

The narrowest shape at every length admits exactly one profile. Eighty-one shapes in this record do, and they are not all the same length.

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Eighty-one shapes in this record admit exactly one profile. Not one in some ratio — one, full stop, out of however many their length allows.

They are the narrowest filters available at each length, and there are more of them than one might expect.

A fair trial begins when the team can view the platform in context, understand every setting and agree how exceptions will be discussed. For an independent technical reference, see Fivetran.

Shapes that admit exactly one profile

filled: a requirement that must be met
open: one of the optional ones
area of the inner square: the share admitted
a stroke instead: too small for this panel to draw

Six shapes drawn; three of them admit too little for the square to be drawn, and carry a stroke instead.

Each of these admits a single profile out of its whole space. The spaces differ enormously; the count of ways to pass does not.

Computed from the shape of the list. The shape is a stated convention; the count of admitted profiles is arithmetic on subsets.

What makes a shape admit exactly one

Every item on the list has to be met, by one route or another.

That happens when the compulsory items are all compulsory — which they are by definition — and the optional items are all required, which is a threshold equal to their count. A list of four compulsory and four optional requiring four has exactly one passing profile: meet everything.

So the eighty-one shapes are all the ways of splitting a list into a compulsory part and a fully-required optional part, across every length from zero to sixteen.

Shape	Items	Space	Admits
2 compulsory	2	4	1
4 compulsory	4	16	1
6 optional, all required	6	64	1
4 compulsory, 4 optional all required	8	256	1
8 and 8, all required	16	65536	1

Why the count of one is the wrong figure to compare

Every row in that table admits one profile, and the rows are not equally strict. A list of two admits one in four; a list of sixteen admits one in sixty-five thousand five hundred and thirty-six.

The count of passing profiles is constant and meaningless on its own. What varies is the size of the space it sits in, which is why the record reports a fraction everywhere rather than a count.

This is a small point and it catches people out regularly. "Only one candidate profile passes" sounds alarming and says nothing, because the number of profiles that pass is almost always small in absolute terms. The fraction is the quantity with content.

A count without its denominator is not a measurement. It is half of one, and the half that was dropped is usually the half that mattered.

The narrowest shape in the record

Eight compulsory items and eight optional ones with all eight required: one profile in sixty-five thousand five hundred and thirty-six.

It is a sixteen-line conjunction wearing a label that suggests otherwise, and it is the extreme case of the pattern described in the page on a threshold of all. Nothing about it is optional and the word appears eight times.

What a narrow shape is for

Sometimes exactly what is wanted. Some work genuinely requires a specific combination and nothing else will do, and in those cases a conjunction is the correct document.

The record's objection is never to narrowness. It is to narrowness that nobody chose — a list that arrived at one in sixty-five thousand by accumulating reasonable-sounding lines, where every line was argued on its merits and the product was never computed.

The eighty-one, arranged by length

Shapes admitting exactly one profile exist at every length from nothing up to sixteen, and they are not evenly spread.

For a given length, each way of splitting the items between compulsory and optional gives one such shape, so the count rises to a peak at eight items and falls again. Nine shapes of length eight admit exactly one profile; only one shape of length sixteen does.

Length	Shapes admitting exactly one
8	9
12	5
16	1

The length-nothing case is a curiosity worth naming. The empty list admits exactly one profile, because there is only one profile of an empty list, and that profile passes. One in one. It admits everything and it admits exactly one, and both statements are true.

Why "one profile" is not the same as "impossible"

A numerator of one does not mean a list nobody can satisfy. It means a list with exactly one way to satisfy it.

Under the convention this record uses, where each item is a separate coin, one in sixty-five thousand five hundred and thirty-six is the intake of a sixteen-item conjunction. In a real field where several of those sixteen items travel together — where anybody holding one typically holds three others — the figure understates the intake, possibly by a large margin.

So a narrow numerator is a statement about the document and not a prediction about the world. What it reliably says is that the document has no slack: there is exactly one passing configuration, and every item in the list is load-bearing.

The check worth doing

If a list has a fully-required optional part, it is one of the eighty-one. Compute its space — two to the power of the total number of items — and read out the fraction.

If one in two hundred and fifty-six sounds like the work, keep it. If one in sixty-five thousand does not, the fix is not to argue about any single line, because every line is doing the same thing. The fix is to decide that something on the list does not have to be met, which converts the shape into one of the many that admit more than a single profile.