ExampleConstruction: AI-generatedVerification: Not suppliedSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-27

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.

Set partitions whose blocks are all singletons have EGF $e^{x}$

Example

There is exactly one partition of $[n]$ into singleton blocks, namely

${{1}, {2}, \dots, {n}} .$

So the counting sequence is constantly $1$ , and the corresponding exponential generating function is

$\sum_{n \geq 0} \frac{x^{n}}{n!} = e^{x} .$

Depends on

The labelled constructions translate into the usual exponential-generating-function rules

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.

Sources

Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics — Symbolic Combinatorics (standard reference, not scraped)

Set partitions whose blocks are all singletons have EGF exex