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

Depends on

• The symmetric group $\operatorname{Sym}(X)$: the bijections of a set $X$ under composition
• The cardinality $\lvert A\rvert$ of a finite set
• Powers $g^{n}$: natural exponents in a monoid and integer exponents in a group, with $g^{0} = e$
• The integers form a commutative ring

Used by

• A parabolic quotient interval of S4 whose Möbius value is 0, so the Eulerian sign formula does not extend to quotients Counterexample
• Two permutations can have the same descent set and different inversion numbers Counterexample
• Column antisymmetrizers, polytabloids, and Specht modules Definition
• 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
• All maximal chains of a rank-three interval in S4, their deleted-position labels, and the lexicographically first chain Example
• All three Specht modules of S₃ Example
• Bruhat versus weak comparability in S4 Example
• Poincare products for Sn, Bn and Dn by explicit insertion Example
• Polytabloids of shape (2,1) Example
• Simple and reflection lengths of a long transposition in S₅ Example
• Subwords, reflection deletions and the covers of the longest element in S4 Example
• The A2 = S3 case: Steinberg inclusion-exclusion, degree product, and reciprocity Example
• The Coxeter complex of A₃: a triangulation of the sphere and the residue of a proper parabolic Example
• The four lifting squares in S4 Example
• The Leibniz formula gives detbeginpmatrixa&bc&d endpmatrix=ad-bc Example
• The Möbius value of the rank-three interval [e,c] in S4 from the recurrence, with the parity and falling-chain checks 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
• Two reduced expressions of one element whose subword descriptions agree Example
• Type-A Artin projections, positive lifts, and the positive braid monoid Example
• Type-A reduced words and inversion numbers in S₃ 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
• Classical Poincare products for A, B, D and I2(m) from the permutation and signed-permutation models Lemma
• Column collision cancels antisymmetrization Lemma
• Composing with a transposition reverses (-1)^inv(σ) Lemma
• Deletion identifies each Specht branching quotient 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
• Ordered and alternating Čech complexes agree Lemma
• Reduced adjacent-transposition words have well-defined positive lifts Lemma
• Standard bases, right actions and bar involutions on the parabolic Hecke modules 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
• Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification Theorem
• The determinant of a triangular matrix is the product of its diagonal entries Theorem

…and 3 more results.

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)