DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-26

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The weak order on $S_{n}$ by inversion-set inclusion

Definition

For $σ, τ \in S_{n}$ , write

$σ \leq_{w e a k} τ : ⟺ Inv (σ) \subseteq Inv (τ),$

where the inversion sets are those of Inversions, inversion number, the sign $sgn (σ) = (- 1)^{inv (σ)}$ , and even and odd permutations.

This is the weak order on $S_{n}$ used later on the combinatorics pathway.

Depends on

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

Used by

The weak and Bruhat orders on S₃ Example

Dependency tree · two levels

5 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

Manytamaris notes, Permutations (standard reference, not scraped)

The weak order on SnSn​ by inversion-set inclusion