RemarkRemark: Literature-sourcedProof: Not applicableSession-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.

This page fixes the Stirling-sign and exponential-generating-function conventions

On this page:

$S (n, k)$ denotes the Stirling numbers of the second kind.
$c (n, k)$ denotes the signless Stirling numbers of the first kind, so $c (n, k)$ counts permutations of $[n]$ with exactly $k$ cycles.
$s (n, k) : = (- 1)^{n - k} c (n, k)$ denotes the signed first-kind numbers used in the inversion formulas.

All exponential generating functions are taken in a commutative $Q$ -algebra, so the coefficients $1 / n!$ and the formal operations $\exp$ and $\log$ of Formal exponential, logarithm, and binomial powers over a commutative $Q$ -algebra are available. The page uses this hypothesis exactly where factorial denominators or formal exponential and logarithmic identities appear.

Used by

Nothing in the library uses this result yet.

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics — Symbolic Combinatorics (standard reference, not scraped)
Herbert S. Wilf, generatingfunctionology, 2nd ed. (standard reference, not scraped)
Andrew Lin, 18.212 Algebraic Combinatorics, Lecture 11: Stirling numbers and more (standard reference, not scraped)