Conjectures of Hopkins, Sagan-Wilson and Defant et al. on Lattices, Parking Functions and the Plactic Monoid
A collection of open problems from the algebraic and enumerative combinatorics literature, resolved in one paper: a conjecture of Defant, Jiang, Marczinzik, Segovia, Speyer, Thomas and Williams on the echelonmotion operator on modular lattices, which also yields a new algebraic bijective proof of Dilworth's theorem; conjectures of Hopkins on parking function statistics studied by Stanley and Yin; and two conjectures of Sagan and Wilson on centralizers in the plactic monoid.
- Result
- Proved
- Status
- Resolved
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Algebraic and enumerative combinatorics
- Posed by
- Colin Defant, Zhongyang Jiang, René Marczinzik, Marco Segovia, David Speyer, Hugh Thomas, Nathan Williams; Sam Hopkins; Bruce Sagan and Jordan Wilson
- Year posed
- —
- Years open
- —
- Solved
- 2026-05-19
- Model
- ChatGPT 5.4 Pro
- Vendor
- OpenAI
- Collaborators
- Colin Defant
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 25 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What the AI did
The claim is unusually flat: all of these proofs were obtained autonomously by ChatGPT 5.4 Pro. The author's role was selecting the problems and writing them up. Worth noting that the first conjecture settled is one the author himself co-posed, so this is a mathematician using a model to close his own open problem.
Verification
arXiv preprint, not peer-reviewed. The proofs are described as short, which makes them checkable by a reader who knows the areas, but no independent verification is recorded and the autonomy claim covers every proof in the paper.
Source
arXiv:2605.19979 - Short Proofs in Algebraic and Enumerative Combinatorics