CounterexampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-26

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Two permutations can have the same descent set and different inversion numbers

Counterexample

The permutations $[1, 0, 2]$ and $[2, 0, 1]$ have the same descent set ${0}$ , but their inversion numbers are $1$ and $2$ respectively.

Facts & Assumptions

Given: Descents and inversion numbers 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.

Verification

technique · direct

1.1

For both permutations, the only descent is at position $0$ : in each case the first entry exceeds the second, while the second does not exceed the third.

given

2.1

The permutation $[1, 0, 2]$ has one inversion, namely $(1, 0)$ , while $[2, 0, 1]$ has two inversions, namely $(2, 0)$ and $(2, 1)$ . So the inversion numbers differ even though the descent sets agree.

step 1.1given∎

Depends on

Used by

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Dependency tree · two levels

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