There are 479 001 600 permutations of twelve things. You would not want the list. But "how many" and "which one is the hundred-millionth" are both perfectly reasonable questions, and neither needs the list to exist.
That is the whole idea here. A combinatorial family is not a sequence you generate and hold — it is a bijection with an interval of the integers. Fix an order, and the family becomes an indexed collection: rank takes an object to its position, unrank takes a position back to an object. Everything else follows.
@enumeratio/collections declares each family as a lazy indexed collection on compute-engine, so its own Count and At answer by arithmetic rather than by enumeration.
12! permutations, and the hundred-millionth of them. Neither call builds a list.
<notatio-cell value="Count(SymmetricGroup(12))" />
<notatio-cell value="SymmetricGroup(12)[100000000]" />
<notatio-cell value="Subsets(20)[700000]" />At is 1-based: At(F, 1) is the first object and At(F, Count(F)) the last. An index outside that range wraps, so a slider can run past the end without erroring.
Ask a family how big it is and the classical numbers fall out — not as a lookup table, but because counting the family is what those numbers do.
Dyck paths are Catalan, set partitions are Bell, k-subsets are binomial. Each pair is the same number arrived at from two directions.
<notatio-cell value="Count(DyckPaths(4))" />
<notatio-cell value="CatalanNumber(4)" />
<notatio-cell value="Count(SetPartitions(4))" />
<notatio-cell value="BellNumber(4)" />
<notatio-cell value="Count(KSubsets(5, 3))" />
<notatio-cell value="Binomial(5, 3)" />This is a good way to check an implementation, and a better way to remember what a counting sequence counts. If Count(Derangements(5)) did not come back 44, one of the two would be wrong.
<notatio-cell value="Count(Derangements(5))" />
<notatio-cell value="Count(Involutions(5))" />
<notatio-cell value="Count(IntegerPartitions(6))" />An index is a dial. Sweep it and you see the family in its order, one object at a time — which is the fastest way to learn what the order is.
Press ▶ and watch the order: the last element climbs slowest. That is colex — subsets are ordered by their largest element first.
<notatio-manipulate params="{ {r, 1}, 1, 56, 1}">
<notatio-figure kind="subset" value="At(KSubsets(8,3), _r)" n="8" />
</notatio-manipulate>The slider's value fills the _r wildcard, the expression is evaluated, and the glyph draws whatever came back. Nothing about that is specific to subsets — point it at a different family and the same rig walks that one.
24 permutations of four things against 14 Dyck paths of semilength four — so the paths wrap around while the permutations are still going. Wrapping is the point: an index is taken modulo the count.
<notatio-manipulate params="{ {r, 1}, 1, 24, 1}">
<notatio-figure kind="permutation" value="At(SymmetricGroup(4), _r)" />
<notatio-figure kind="dyck" value="At(DyckPaths(4), _r)" />
</notatio-manipulate>Once an object is addressable, a statistic of it is an ordinary function call. The permutation statistics — inversions, descents, major index, cycles — compose with At directly, which makes "what does this statistic look like across the family" a question you can just ask.
The 7th permutation of four things is [2,1,3,4] — one inversion, one descent.
<notatio-cell value="SymmetricGroup(4)[7]" />
<notatio-cell value="Inversions(SymmetricGroup(4)[7])" />
<notatio-cell value="Descents(SymmetricGroup(4)[7])" />A family and an order are not separable: change the order and At answers differently, so the order has to be stated. KSubsets is colexicographic — the 4th 3-subset of {1..5} is {2,3,4}, not {1,2,5}, because colex orders by the largest element first.
First and last. Colex is what makes the rank formula a sum of binomials — which is exactly the combinatorial number system.
<notatio-cell value="KSubsets(5, 3)[1]" />
<notatio-cell value="KSubsets(5, 3)[10]" />Unranking is the same operation as writing a number in a numeral system whose places are combinatorial rather than geometric — the factoradic digits of
Every family here is catalogued, with its order and its counting sequence, in the symbol reference. The glyphs are <notatio-figure> and the slider is <notatio-manipulate>.