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Counting fixed-point-free Cayley permutations

Giulio Cerbai, Anders Claesson

TL;DR

The work develops a two-sort species framework to count fixed-point-free Cayley permutations (Cayley-derangements) by embedding Cayley functional digraphs into $R$-recurrent structures and deriving a differential equation for $oldsymbol{ extPsi}_R$. It yields an explicit formula for Cayley-derangements in terms of $r$-Stirling numbers and subfactorials, and demonstrates multiple equivalent counting expressions for End$_R$ and Cay$_R$, including unisort and two-sort formulations. A two-sort extension of Joyal's bijection connects Cayley permutations with rooted-tree layouts, while a unisort product approach provides compact identities for Cayley-recurrent structures across trees, forests, and connected digraphs. The authors also investigate statistics and asymptotics, conjecturing that Cayley-derangements share the universal $1/e$ fixed-point-free limit, and outline broad generalizations via $oldsymbol{ extPsi}_{R,T}$ and other tree-classes with potential applications to broader classes of functional 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 conjecture 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.

Counting fixed-point-free Cayley permutations

TL;DR

The work develops a two-sort species framework to count fixed-point-free Cayley permutations (Cayley-derangements) by embedding Cayley functional digraphs into -recurrent structures and deriving a differential equation for . It yields an explicit formula for Cayley-derangements in terms of -Stirling numbers and subfactorials, and demonstrates multiple equivalent counting expressions for End and Cay, including unisort and two-sort formulations. A two-sort extension of Joyal's bijection connects Cayley permutations with rooted-tree layouts, while a unisort product approach provides compact identities for Cayley-recurrent structures across trees, forests, and connected digraphs. The authors also investigate statistics and asymptotics, conjecturing that Cayley-derangements share the universal fixed-point-free limit, and outline broad generalizations via and other tree-classes with potential applications to broader classes of functional 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 conjecture that their proportion tends to , as for permutations and endofunctions. Our approach also yields counting formulas when the functional digraph is a tree, forest, or connected.

Paper Structure

This paper contains 9 sections, 11 theorems, 96 equations, 6 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Combinatorial species and functional digraphs
  3. Cayley permutations
  4. R-recurrent functional digraphs
  5. A differential equation
  6. A two-sort species extension of Joyal's bijection
  7. Unisort species of Cayley functional digraphs
  8. Statistics and asymptotics
  9. Generalizations

Key Result

Lemma 3.1

Let $f\in\mathrm{End}[n]$. Then $f$ is a Cayley permutation with $\mathrm{Im}(f)=[k]$ if and only if the internal nodes in the functional digraph of $f$ have labels $[k]$.

Figures (6)

  • Figure 1: The functional digraph of $f=985776326459548\in\mathrm{End}[15]$, where the $i$th letter of $f$ is $f(i)$. Internal nodes are black while leaves are white.
  • Figure 2: The functional digraph of $f=693163933\in \mathrm{End}[9]$, on the left, and the corresponding rooted tree, on the right.
  • Figure 3: The doubly-rooted tree associated with the endofunction of Figure \ref{['figure_endofun']} under Joyal's bijection. The spine is colored in blue; tail and head are distinguished with one and two circles, respectively.
  • Figure 4: An $R$-recurrent functional digraph of two sorts. Nodes of sort $X$ (internal nodes) are black and labeled by numbers; nodes of sort $Y$ (leaves) are white and labeled by letters.
  • Figure 5: The construction manifested in equation \ref{['eqdiff_psiR']}.
  • ...and 1 more figures

Theorems & Definitions (21)

  • Lemma 3.1
  • Definition 4.1
  • Definition 4.2
  • Theorem 5.1
  • proof
  • Lemma 5.2
  • proof
  • Theorem 5.3
  • proof
  • Corollary 5.4
  • ...and 11 more