DefinitionDefinition: 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.

The signed and signless Stirling numbers of the first kind

Definition

For $n, k \in N$ , let $S (n, k)$ be the set of permutations of $[n]$ with exactly $k$ cycles in the sense of Support, fixed points, disjoint cycles, cycle length, disjoint-cycle decompositions, and cycle type. This is a finite set because $S_{n}$ is finite by The Lehmer code gives $∣ S_{n} ∣ = n!$ again.

The signless Stirling number of the first kind is

$c (n, k) : = ∣ S (n, k) ∣ .$

The signed Stirling number of the first kind is

$s (n, k) : = (- 1)^{n - k} c (n, k) .$

In particular, $c (0, 0) = 1$ from the identity permutation of $[0] = \emptyset$ , and $c (n, k) = 0$ when $k > n$ or when $n \geq 1$ and $k = 0$ .

Depends on

Used by

Dependency tree · two levels

7 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

Andrew Lin, 18.212 Algebraic Combinatorics, Lecture 11 (standard reference, not scraped)
Herbert S. Wilf, generatingfunctionology, 2nd ed., §3.5 (standard reference, not scraped)