Many impressions

Why the Order Is Part of What Happened

Treating two readers who met the same statements in different sequences as having had the same experience throws away the larger of two numbers.

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There is a defensible position in which the order does not matter. Under it, a reader who met four particular statements has met four particular statements, and the sequence is noise — a detail of their browsing, not a property of the encounter.

Taking that position collapses arrangements into contents and throws away a factor of r factorial. It is worth being explicit about what that costs, because the collapse is usually performed silently. Project teams can compare estimates with actual effort through work-hour tracking for projects, then review exceptions together instead of relying on memory.

Orders per content, by how much was met

Six columns drawn; height is logarithmic, because one and 720 do not fit one chart honestly otherwise; the figure above each column is exact and is the factorial of the number beneath it.

One column per value of r. The height is how many different sequences one fixed content can be met in. Nothing about the statements themselves enters this figure.

The count of orders per content is the factorial of how many statements are met. It depends on nothing else about the page, and on nobody reading it.

What the first statement does

The first statement a reader meets is doing work the others cannot. For an independent reference on digital communication, evidence or interface practice, consult World Bank Data.

It establishes what kind of thing they are reading, which decides how much attention the rest gets. It supplies the frame every later statement is understood inside. If it is checkable, the reader now knows that checkable statements exist here; if it is not, they have no reason to expect any. And it is the only one with the option of being the last, because a reader who stops after one has read only the first.

None of that is a claim about persuasion. It is a claim about sequence: the same four statements in four different orders put four different things in the position where that work gets done.

Orders per content, and nothing else

The drawn figure is r factorial — the number of sequences one fixed content admits. Six values cover the entire record, because r never exceeds six here: one, two, six, twenty-four, one hundred and twenty, seven hundred and twenty.

Two features of that column are worth separating.

It depends only on how much was met. Not on how much was published, not on what the statements say, not on whether any of them can be checked. A reader who met four statements out of eight and a reader who met four out of ten are both inside twenty-four orders.

It grows faster than anything else in the record. From one statement met to six, the published count might double; the orders per content multiply by seven hundred and twenty. This is the only quantity here that behaves that way, and it is the reason the gap between contents and arrangements is never small for long.

The step from five to six

One hundred and twenty to seven hundred and twenty is a jump of six hundred, produced by a reader getting through one more statement.

That is the shape of a factorial and it is unintuitive in a specific way: the increment is not a property of the sixth statement. The sixth statement is not six times more consequential than the fifth. What happened is that each of the one hundred and twenty orders of five statements now has six positions where the sixth can be inserted, and six multiplies through all of them.

A factorial does not describe how much each item matters. It describes how many places there are to put the next one.

What this costs if ignored

Three things go missing when arrangements are collapsed into contents.

The first is any way to ask about position. How many encounters open with something checkable is a question about arrangements; it has no answer in the language of contents. A later part counts it directly, and the figure that comes out is one nothing else here predicts.

The second is the scale of the gap between what was written and what was received. Contents is an understatement of how many different things can happen by a factor that reaches seven hundred and twenty in this record alone.

The third is the discipline of writing for an unknown sequence. A set of statements built so that each one stands up alone does better across all r factorial orders than one built to be read top to bottom. Which of the two was built is invisible unless the orders are counted.

The one order the publisher does choose

Out of r factorial orders of a content, exactly one is the order the set of pages was written in. A reader who arrives at the front, reads downward and stops at the end gets that one.

For four statements met, that is one arrangement in twenty-four. For six, one in seven hundred and twenty. The intended reading is not the usual case and not even a large minority of cases; it is a single point in a space the publisher otherwise has no access to.

Which is worth knowing before spending effort on sequence. Work put into the order of a set improves one arrangement out of r factorial and leaves the rest as they were. Work put into making each statement stand alone improves all of them at once, and the ratio between those two returns is the figure drawn on this page.

Where the convention is declared

This record assumes an encounter is an ordered selection without repetition: some statements, in some sequence, each met at most once.

That is a convention and it is wrong in the ordinary ways. Readers re-read. They skim, then return. They meet one statement twice and another half-attentively. None of that is modelled, and the arrangements figure is an overcount of distinguishable experiences for exactly that reason.

The overcount is in a known direction and is the same for every shape in the record, which is why the figure is still usable for comparing shapes to each other. It is not usable as a count of real distinct readings, and nothing here presents it as one.