Telling apart

Four Hundred And Sixty-One Schemas

How many structures fit inside the bounds, how they spread across one to six facets, and why the order of filters is not part of a schema.

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Facets carry two to six values. A schema carries one to six facets. Count the distinct structures inside those bounds and the answer is four hundred and sixty-one.

Facets	Schemas
one	5
two	15
three	35
four	70
five	126
six	210

Each row is the number of ways to choose that many sizes from five available, with repeats allowed and order disregarded.

When a faceted model becomes part of a real operating dashboard, teams evaluating employee monitoring software should define the decision, access rules and correction path before collecting any signals. For a separate technical reference, see NumPy.

Why order is disregarded

Because moving a dropdown up the sidebar changes nothing this record measures.

A board with filters of four, six and six choices has one hundred and forty-four cells whichever order they appear in. The questions it permits number two hundred and forty-five in every arrangement. The reachable result sizes are the same set.

So a schema here is an unordered collection: four, six and six is one schema, not six schemas. Counting arrangements separately would inflate the record without adding a single distinct structure to it, and every figure would carry a factor that means nothing.

Order affects how a board reads and how quickly anybody finds the control they want. It does not affect what the board can tell apart or what it can be asked, and this record measures only those.

Why repeats are allowed

Because two filters of six choices is an ordinary thing for a board to have, and a record that excluded it would be describing something other than boards.

Four of the seven schemas in the largest class contain a repeated size. The schema with the most cells in the whole record is six filters of six. Repeats are not an edge case; they are most of the territory.

The shape of the distribution

More than half the schemas carry five or six facets. Two hundred and ten of the four hundred and sixty-one have six, and one hundred and twenty-six have five: three hundred and thirty-six between them.

That is a fact about counting rather than about boards. There are simply more ways to pick six sizes than to pick two, so the long schemas dominate the enumeration. It does not mean most real boards carry six filters, and no figure in this record should be read as a claim about what is common.

This matters whenever a figure here gets quoted as though it described the world. The median cell count across the record is eight hundred and sixty-four — but that is the median of an enumeration that leans heavily towards long schemas, not the median of any collection of actual boards.

What the bounds leave out

Three things, each covered properly in the section on limits.

Facets with more than six values. Real boards carry filters with dozens of options. The arithmetic is unchanged — multiply — but the figures leave this record's tables.

More than six facets. Same answer: the method carries on, the tables stop.

Multiple selection. A board letting a reader tick several values at once in the same filter is doing different arithmetic, and the difference is large rather than marginal.

Of the three, the last alters the structure rather than merely the scale, and that is why a whole page is given over to it.

The two hundred and ten

Six-facet schemas are the largest group in the enumeration and the one most worth being careful about.

Two hundred and ten of them exist, which is nearly half the record. They have the lowest shares of pinning questions, the heaviest grain collapse, and the largest cell counts. Any median computed across the record is pulled towards them.

That is why every median here is labelled as a median of the enumeration rather than of anything in the world. A collection of real boards would not be half six-facet structures, and a figure quoted without that qualification would be a claim about boards that the record has no basis for.

What the four hundred and sixty-one are for

They are a range to place a board inside, not a catalogue of possibilities.

Nobody needs to know that four hundred and sixty-one structures exist in order to compute their own board's figures: multiplying the sizes takes a minute and needs no table. The enumeration earns its keep differently — it is what makes the statements about collision possible.

Without counting the whole range, "cell counts collide often" is an impression. With it, the statement becomes exact: three hundred and eighty-one schemas of four hundred and sixty-one share a cell count, one hundred and fifteen groups do the sharing, the largest holds seven, and no two schemas anywhere share both figures.

Those four statements are the record. The enumeration exists to make them checkable rather than plausible.