DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-11

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.

Inversions, inversion number, the sign $sgn (σ) = (- 1)^{inv (σ)}$ , and even and odd permutations

Definition

Let $n \in N$ and $σ \in S_{n} = Sym (n)$ . An inversion of $σ$ is a pair $(i, j)$ with $i < j < n$ and $σ (i) > σ (j)$ . The inversion set and inversion number are

$Inv (σ) : = {(i, j) \in n \times n : i < j and σ (i) > σ (j)}, inv (σ) : = ∣ Inv (σ) ∣ .$

The sign of $σ$ is the integer

$sgn (σ) : = (- 1)^{inv (σ)} \in {+ 1, - 1} .$

The permutation is even when its sign is $+ 1$ , equivalently when its inversion number is even, and odd when its sign is $- 1$ , equivalently when its inversion number is odd. For $n = 0$ or $n = 1$ , every inversion set is empty, so the unique permutation is even.

Depends on

Dependency tree · 2 levels

Sources

Stanford Math 51H, Permutations

Inversions, inversion number, the sign sgn⁡(σ)=(−1)inv⁡(σ)sgn(σ)=(−1)inv(σ), and even and odd permutations

Definition

Depends on

Dependency tree · 2 levels

Sources

Inversions, inversion number, the sign $sgn (σ) = (- 1)^{inv (σ)}$ , and even and odd permutations