enumeratiocombinatorial collections, where the math is data

The name ​

Latin ēnumerātiō (f.), from ēnumerāre "to count out, reckon up" (ē-/ex- "out" + numerāre "to count," from numerus "number"). In classical rhetoric, the enumeratio is the closing recapitulation that lists the points made.

The name holds both halves of the project's lineage. Numeration — writing a number positionally — and enumeration — counting and listing the structures that organize those writings — share the root numerus. enumeratio began as a study of number systems and grew into a general, database-driven library of the combinatorial structures enumeration is about — permutations, set partitions, compositions, subsets, Dyck paths, integer partitions, … — where the math is data.

The model — the catalog is the database ​

A collection is a row in the catalog (base_collection + its grade axes, orders, and examples) plus a carrier type and a handful of engine functions. A generator (base_realize) turns each collection row into the generic layer, so adding a collection is rows + its engine, never per-collection glue.

• handle addressing — permutations(5) is a subscriptable handle. Grades bind at construction (set_partitions(5, blocks => 2)) and grading is multi-axis; a subscript addresses the element by rank or by a named order. cardinality(handle) is ∞-aware.
• membership is data — x <@ collection / collection @> x dispatch to a generated contains(handle, x); a collection's is_* predicate folds in as its membership_predicate, and grade axes are checked for free.
• stat = a per-element invariant; map = a function to another collection; restrictions name a slice as its own collection.
• representations are data — notation and figures: render() casts to a chosen medium, and glyph_svg(<carrier>) emits an element's SVG straight from the db.
• examples are data — the example suite runs every catalog example; they double as regression tests and documentation, and COMMENT ON makes the DB self-describing.

One identity, many roles ​

What most combinatorics libraries miss is the connections. A combinatorial number is written once — as a math_* identity (math_factorial, math_binomial, math_stirling2, …) — and every role that is that number references it instead of re-deriving it: the cardinality of a collection (one Catalan identity is the count of Dyck paths, binary trees, ordered trees, 231-avoiding permutations, and non-crossing partitions alike); an aggregate's closed form (the Stirling-2 row that the grade counts of set_partitions(5) sum to Bell(5)); and — the deep one — the order-isomorphic maps between equinumerous families, which let one borrow another's ranking (binary_trees ranks through the Dyck-path bijection, never writing its own).

Getting around ​

• Guide — the docs site itself, and the live, in-browser components.
• Atlas and Math — the collections and their geometry.
• Packages — pg (the SQL core), client, cli, components.
• Design — the model, the spikes, the roadmap.
• Explorer — browse it all interactively.