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 sequence construction $SEQ (A)$

Definition

Let $A$ be a combinatorial class. Its sequence construction $SEQ (A)$ is the class of all finite ordered sequences

$σ = (a_{1}, \dots, a_{r}) (r \geq 0, a_{i} \in A),$

with size

$∣ σ ∣ : = ∣ a_{1} ∣ + \dots + ∣ a_{r} ∣ .$

The case $r = 0$ is the empty sequence, whose size is $0$ ; it is the unique object of the neutral class $E$ .

Write ${SEQ}_{\geq 1} (A)$ for the subclass of nonempty sequences.

If $A$ has an object of size $0$ , then $SEQ (A)$ need not be a combinatorial class: infinitely many different lengths can produce the same total size. The generating-function theorem therefore carries a no-size-zero hypothesis.

Depends on

Used by

Dependency tree · two levels

5 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 sequence construction SEQ⁡(A)SEQ(A)