The Proportion of Permutations Fixing a k-Set
An asymptotic formula for , the limiting probability that a random permutation has an invariant set of size : it is asymptotically times a smooth positive function, sharpening a line of estimates running through Łuczak-Pyber and Eberhard-Ford-Green.
- Result
- Proved
- Status
- Resolved
- AI contribution
- AI co-developed
- Method
- Argument
- Field
- Permutation combinatorics
- Posed by
- —
- Year posed
- —
- Years open
- —
- Solved
- 2026-04-30
- Model
- GPT o4-mini, ChatGPT Pro 5.4
- Vendor
- OpenAI
- Collaborators
- Ben Green, Mehtaab Sawhney
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 15 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What the AI did
The authors "originally discovered that g was almost constant via the use of GPT o4-mini. Upon being asked to prove this fact, the model suggested the broad strategy" (initially miscomputing an integral, then completing the evaluation after being pointed at Mellin transforms). A separate argument in the paper is ChatGPT Pro 5.4's, shortened and checked by the authors, kept because it was cleaner than their own.