False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-26

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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.

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FALSE: the major index equals the inversion number for every permutation

Statement

False claim: for every permutation $σ$ , one has

$maj (σ) = inv (σ) .$

What is true is the weaker distributional statement of The major-index generating function of $S_{n}$ is $[n]_{q}!$ .

Facts & Assumptions

Given: Major index and inversion number are defined by Descent sets, major index, excedances and fixed points of a permutation and Inversions, inversion number, the sign $sgn (σ) = (- 1)^{inv (σ)}$ , and even and odd permutations.

Refutation

technique · direct

1.1

Take $σ = [2, 0, 1]$ . Its only descent is at position $0$ , so $maj (σ) = 1$ .

given

2.1

The same permutation has two inversions, namely $(2, 0)$ and $(2, 1)$ , so $inv (σ) = 2$ . Therefore $maj (σ) \neq inv (σ)$ , and the claim is false.

step 1.1given∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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

Richard P. Stanley, Enumerative Combinatorics, Volume 1, second edition (standard reference, not scraped)