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combinatorics / Algebraic combinatorics

Stanley's Claw-Free Schur-Positivity Conjecture

Is the chromatic symmetric function $X_G$ Schur positive for every claw-free graph $G$? Two explicit $12$-vertex line graphs have Schur coefficients $-64$ and $-40$ at $s_{(3,3,3,3)}$.

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25Significance / 100
2Frontier events
0Verification tasks
1Recorded attempts

Temporal state

Current frontier

Independently reproduced

Both published negative Schur coefficients independently reproduced exactly

Append-only history

Frontier timeline

combinatoricsAug 29, 2026Significance 78/100

Both claw-free Schur counterexample coefficients independently reproduced

Prior state unknown→Both published negative Schur coefficients independently reproduced exactly

An independent exact enumerator rebuilt both line graphs from the published edge lists and obtained the stated coefficients −64 and −40. VibeMathed's external label remains unreviewed; this is a separate local verification event.

Independently reproducedStanley's Claw-Free Schur-Positivity Conjecture →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review
combinatoricsJul 23, 2026Significance 25/100Registry: unreviewed

Stanley's Claw-Free Schur-Positivity Conjecture

Prior state unknown→disproved

Is the chromatic symmetric function $X_G$ Schur positive for every claw-free graph $G$? Two explicit $12$-vertex line graphs have Schur coefficients $-64$ and $-40$ at $s_{(3,3,3,3)}$.

Source authenticatedStanley's Claw-Free Schur-Positivity Conjecture →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Research memory

Claims and attempts

Scoped claims

Independently reproduced

For the line graphs G₁ and G₂ defined by the paper's two 12-edge source graphs, the coefficient of s_(3,3,3,3) in the chromatic symmetric function is exactly −64 and −40, respectively.

The edge lists are transcribed from arXiv:2607.21508v2. Line graphs are claw-free.

Source authenticated

Is the chromatic symmetric function $X_G$ Schur positive for every claw-free graph $G$? Two explicit $12$-vertex line graphs have Schur coefficients $-64$ and $-40$ at $s_{(3,3,3,3)}$.

Recorded attempts

Recompute both Schur coefficients from the published edge lists

success · MFN independent verifier

The 22-edge graph returned −64 and the 21-edge graph returned −40 exactly.

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