CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-26

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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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

1.1given

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 $2$ .

2.1step 1.1given∎

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.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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