Counting fixed-point-free Cayley permutations

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 slash e $1/e$ <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mi>e</mml:mi> </mml:mrow> </mml:math> , as for permutations and endofunctions. Our approach also yields counting formulas when the functional digraph is a tree, forest, or connected.