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Counting fixed-point-free Cayley permutations
Published online by Cambridge University Press: 28 May 2026
Abstract
Two-sort species yield differential equations for functional digraphs of Cayley permutations. From these, we obtain an explicit formula for fixed-point-free Cayley permutations and prove that their proportion tends to 
$1/e$, as for permutations and endofunctions. Our approach also yields counting formulas when the functional digraph is a tree, forest, or connected.
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- © The Author(s), 2026. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.
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