The Proportion of Permutations Fixing a k-Set
Statement
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.
Context
The invariant-set proportion is a documented question of permutation combinatorics with a real literature from Łuczak-Pyber through Eberhard-Ford-Green.
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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.
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