Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Plane trees with at most four vertices from
Example
The rooted plane trees on at most four vertices are counted by the first coefficients of the Catalan series:
Concretely there is one tree on one vertex, one on two vertices, two on three vertices, and five on four vertices.
Facts & Assumptions
Given: The plane-tree generating function satisfies (Rooted plane trees satisfy ).
Verification
Solving coefficientwise gives , so the first four counts are .
These are realized by the evident shapes: a single root; a root with one child; for three vertices, the chain and the root with two children; for four vertices, the chain of length four, the root with three children, the root with one child whose child has two children, and the two left-right orderings of a root with two children one of which has one child.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.