Counting fixed-point-free Cayley permutations

TL;DR

This paper derives explicit formulas for fixed-point-free Cayley permutations using species and differential equations, showing their proportion approaches 1/e, similar to permutations and endofunctions, and extends to related structures.

Contribution

It introduces a novel species-based differential equation approach to count fixed-point-free Cayley permutations and related structures, providing explicit formulas and asymptotic proportions.

Findings

Proportion of fixed-point-free Cayley permutations tends to 1/e.

Explicit counting formulas for permutations, endofunctions, and related structures.

Method extends to trees, forests, and connected digraphs.

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.