CounterexampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge 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 number of excedances is Eulerian but not Mahonian

Counterexample

The statistic $exc$ is Eulerian, because it is equidistributed with descents, but it is not Mahonian: on $S_{3}$ its distribution is $1, 4, 1$ , while the inversion distribution is $1, 2, 2, 1$ .

Facts & Assumptions

Given: Descents and excedances are equidistributed (Descents and excedances are equidistributed on $S_{n}$ ), while the major-index generating function is $[n]_{q}!$ and so agrees with the inversion distribution (The major-index generating function of $S_{n}$ is $[n]_{q}!$ ).

Verification

technique · direct

1.1

On $S_{3}$ , the excedance counts are distributed as $1, 4, 1$ : there is one permutation with $0$ excedances, four with $1$ , and one with $2$ .

given

2.1

On $S_{3}$ , the inversion numbers are distributed as $1, 2, 2, 1$ over $0, 1, 2, 3$ . Since these two distributions are different, $exc$ is not Mahonian.

step 1.1given∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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.