Alphabeta MathHow 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.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Inversions, inversion number, the sign , and even and odd permutations
Definition
Let and . An inversion of is a pair with and . The inversion set and inversion number are
The sign of is the integer
The permutation is even when its sign is , equivalently when its inversion number is even, and odd when its sign is , equivalently when its inversion number is odd. For or , every inversion set is empty, so the unique permutation is even.
Depends on
Used by
- Two permutations can have the same descent set and different inversion numbers Counterexample
- For n≥1, the determinant over a commutative ring by the Leibniz formula, and |det A| for a real matrix Definition
- The weak order on Sₙ by inversion-set inclusion Definition
- A full 3×3 Leibniz expansion lists all six permutations and their signs Example
- The Leibniz formula gives detbeginpmatrixa&bc&d endpmatrix=ad-bc Example
- The six elements of S₃: one-line form, cycle structure, inversions, and sign Example
- Two different transposition factorisations of the same permutation have the same parity Example
- FALSE: the major index equals the inversion number for every permutation False statement
- A cyclic permutation of the coordinates of ℝ³ preserves Jordan measurability and integrals Lemma
- Composing with a transposition reverses (-1)^inv(σ) Lemma
- Each coordinate of the oriented area vector is the Jacobian determinant of the matching cyclic projection Lemma
- Every alternating multilinear F satisfies F(A)=F(I)∑_σ∈ Sₙsgn(σ)∏ᵢ a_σ(i),i Lemma
- Tail-swapping is a sign-reversing involution on the intersecting systems Lemma
- The Vandermonde product transforms by the sign of the root permutation Proposition
- This page uses the published inversion number and the shifted Eulerian indexing A(n,k) Remark
- Foata's transformation sends major index to inversion number Theorem
- The determinant of a triangular matrix is the product of its diagonal entries Theorem
- The inversion generating function of Sₙ is [n]_q! Theorem
- The sign is a homomorphism Sₙ→{+1,-1}, surjective exactly when n≥ 2 Theorem
Dependency tree · two levels
32 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
- Stanford Math 51H, Permutations (standard reference, not scraped)