DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-26

How 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.

The neutral class $E$ and the atomic class $Z$

Definition

The neutral class $E$ has one object $ϵ$ of size $0$ and no objects of any other size. Its counting sequence is $1, 0, 0, \dots$ , so its ordinary generating function is $1$ .

The atomic class $Z$ has one object $z$ of size $1$ and no objects of any other size. Its counting sequence is $0, 1, 0, 0, \dots$ , so its ordinary generating function is $x$ .

Both are combinatorial classes in the sense of Combinatorial classes, counting sequences and ordinary generating functions.

Depends on

Combinatorial classes, counting sequences and ordinary generating functions

Used by

Binary words have generating function 1/(1-2x) Corollary
Positive-integer compositions have generating function x/(1-2x) Corollary
Rooted plane binary trees satisfy B(x)=1+xB(x)² Corollary
Rooted plane trees satisfy T(x)=x/(1-T(x)) Corollary
The number of necklaces of length n on an m-letter alphabet is 1/n∑_d∣ nφ(d)m^n/d Corollary
The sequence construction SEQ(A) Definition

Dependency tree · two levels

4 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.

Sources

Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics (standard reference, not scraped)
Robert Sedgewick and Kevin Wayne, Analysis of Algorithms, Section 3.9 (standard reference, not scraped)

The neutral class EE and the atomic class ZZ