Length

How Many Shapes Each Length Allows

Forty-five distinct filters can be built from eight lines. The count of shapes at each length, where it peaks, and what it says about reading a list.

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Eight lines on a page can be arranged into forty-five different filters within this record's bounds. Six lines into twenty-eight. Two lines into six.

Length	Distinct shapes
0	1
2	6
4	15
6	28
8	45
10	42
12	35
14	24
16	9

Four hundred and five shapes in total, peaking at length eight.

Where the count comes from

For a list of length L, the compulsory part can be any size from none up to L, provided the optional remainder does not exceed eight. For each split, the threshold can be any whole number from none up to the size of the optional part.

The next useful step is to compare what the service currently offers, choose one live automation cycle and compare the report with what was delivered. For an independent technical reference, see Twilio Segment.

So the count is a sum over splits of one more than the optional count — which produces the rise to forty-five and then the fall as the record's bound starts removing splits.

What forty-five means for a reader

It means that a reader holding an eight-line list, with no rule stated, is looking at one of forty-five possible filters and has no way to tell which.

Those forty-five run from admitting everything to admitting one profile in two hundred and fifty-six. Picking among them by reading the tone of the document is not a method; it is a guess with forty-five outcomes.

A list without its rule is not an under-specified filter. It is forty-five different filters printed on the same sheet of paper, and the reader chooses one without knowing they are choosing.

Why eight is the peak

Because that is where the record's two constraints cross. Below eight, longer lists allow more splits and the count rises. Above eight, the cap on the optional part starts forbidding splits and the count falls.

The peak is therefore a feature of the bounds rather than of lists. Without the cap the count would keep rising with length, which is the more realistic picture and also the one that makes the reader's position worse: longer real lists are chosen from even larger sets of possible filters than this table shows.

The one shape per length that needs no rule

There is an exception worth naming. A list with no optional items at all — a pure conjunction — has exactly one shape at its length, and its intake follows from the halving law alone.

That is why a pure conjunction is the only kind of list that can be read correctly without a stated rule. It is also, as the page on the unwritten rule argues, the reading that gets applied by default to lists which were not meant that way.

The count at other lengths

Forty-five is the peak. The count rises to it and comes back down, and it does so asymmetrically because the two parts of a list are capped separately.

Length	Distinct shapes
0	1
4	15
7	36
8	45
9	44
16	9

Adding the nine-item column: the count has barely fallen, because the only shape lost is the one with no compulsory items at all. From there it drops steadily, and at sixteen items only nine shapes remain — one for each threshold over the eight optional items, with the compulsory part pinned at eight.

Summed across all seventeen lengths this gives the four hundred and five shapes the record holds.

Why forty-five and not nine

A common mistake is to count thresholds instead of shapes.

Eight lines admit nine thresholds if all eight lines are optional. But the eight lines can also be split differently — seven optional and one compulsory, six and two, and so on down to none optional — and each split carries its own set of thresholds.

Adding those up gives forty-five: one shape with no optional items, two with one, three with two, and so on to nine with eight. The sum of the first nine whole numbers, which is why the peak falls where it does.

The shapes inside the count that nobody writes

Not all forty-five are documents anybody would produce, and the count does not pretend otherwise.

Several are degenerate. One has eight optional items and a threshold of none, filtering nothing. One has eight optional items and a threshold of all eight, which is a conjunction in disguise. A handful sit so close to those two that the same objection applies.

They are counted because the enumeration is mechanical and a mechanical enumeration is checkable. What matters is not that all forty-five are plausible but that the plausible ones are among them, and that no shape has been quietly left out because it looked unlikely to whoever was counting.

What this adds up to

The number of lines on a list is a weak signal and the number of possible filters behind those lines is large. The rule is what collapses forty-five possibilities to one.

Everything in this section says a version of that. The next two pages take the practical consequences: what a short list can do that a long one cannot, and what happens when somebody writes a list to a target length rather than to a target filter.