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

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 Bruhat order on $S_{n}$ by rank inequalities

Definition

For $σ \in S_{n}$ and $0 \leq p, q < n$ , define the rank number

$r_{σ} (p, q) : = ∣ {i \in n : i \leq p and σ (i) \leq q} ∣ .$

For $σ, τ \in S_{n}$ , write

$σ \leq_{B r u h a t} τ : ⟺ r_{σ} (p, q) \geq r_{τ} (p, q) for every 0 \leq p, q < n .$

This is the rank-inequality form of the Bruhat order.

Depends on

The finite symmetric group $S_n$, one-line notation, and cycle notation

Used by

The weak and Bruhat orders on S₃ Example

Dependency tree · two levels

3 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

Arun Ram, Notes on Schubert Polynomials, Chapter 1: Permutations (standard reference, not scraped)

The Bruhat order on SnSn​ by rank inequalities