Fixed-Point-Free Cayley Permutations

• Fixed-point-free Cayley permutations are endofunction models where no element maps to itself, defined via a packed-word constraint and loop-free digraph structure.
• Their enumeration employs a two-sort species approach with difference r-Stirling numbers, leading to explicit formulas and open asymptotic questions.
• They connect diverse combinatorial frameworks including surjective-word combinatorics, insertion encodings, and cycle-parity methods, inspiring further research on fixed-point phenomena.

Fixed-point-free Cayley permutations are Cayley permutations in the packed-word or endofunction sense that have no classical fixed points. Concretely, a Cayley permutation on $[n]$ is a function $f:[n]\to[n]$ with $\operatorname{Im}(f)=[k]$ for some $k\le n$, equivalently a word $f(1)\cdots f(n)$ whose set of distinct letters is an initial segment of the positive integers; it is fixed-point-free if $f(i)\neq i$ for all $i$ (Cerbai et al., 12 Jul 2025). In the functional-digraph model this means precisely that the associated digraph has no loops. The subject sits at the intersection of surjective-word combinatorics, endofunction digraphs, species theory, and several adjacent literatures in which “fixed point” may instead refer to operator fixed points or to the absence of 1-cycles in ordinary permutations with parity-constrained cycle structure (Cerbai et al., 12 Jul 2025).

1. Definitions, equivalent models, and scope

A Cayley permutation on $[n]$ is an intermediate object between a permutation and an arbitrary endofunction: every permutation has image $[n]$, hence is a Cayley permutation, and every Cayley permutation is an endofunction. In word notation, it is a finite word $w_1\cdots w_n$ such that if a letter $f:[n]\to[n]$0 appears, then every positive integer $f:[n]\to[n]$1 also appears (Cerbai et al., 12 Jul 2025). The species-theoretic literature identifies the same class with ballots, or ordered set partitions, via the map $f:[n]\to[n]$2; this yields the exponential generating function

$f:[n]\to[n]$3

so the total number of Cayley permutations of length $f:[n]\to[n]$4 is the $f:[n]\to[n]$5-th Fubini number (Claesson et al., 2024).

Within this model, the natural classical notion of a fixed point is positional: $f:[n]\to[n]$6 is a fixed point if $f:[n]\to[n]$7. A fixed-point-free Cayley permutation, also called a Cayley-derangement, is therefore a Cayley permutation with no such indices (Cerbai et al., 12 Jul 2025). This notion is fundamentally different from pattern avoidance in the usual Cayley-permutation sense, because pattern containment only records relative order and equalities among letters, whereas the condition $f:[n]\to[n]$8 couples an absolute position with an absolute value. As a result, fixed-point-freeness is not expressible as avoidance of a single classical Cayley pattern and is not itself a Cayley permutation class $f:[n]\to[n]$9 closed under taking patterns (Claesson et al., 2024).

Framework	Underlying objects	Meaning of “fixed-point-free”
Packed-word / endofunction model	Cayley permutations $\operatorname{Im}(f)=[k]$0 with $\operatorname{Im}(f)=[k]$1	$\operatorname{Im}(f)=[k]$2 for all $\operatorname{Im}(f)=[k]$3
Stack-sorting model	Cayley permutations acted on by $\operatorname{Im}(f)=[k]$4	operator-fixed-point-free: $\operatorname{Im}(f)=[k]$5
Cayley-continuant parity model	Ordinary permutations in $\operatorname{Im}(f)=[k]$6	all cycles are even, hence no 1-cycles

The table records three nearby usages that appear in the literature (Cerbai, 2020, Chen, 2023).

2. Functional digraphs and the structural characterization

The most effective structural model is the functional digraph of an endofunction $\operatorname{Im}(f)=[k]$7: vertices are $\operatorname{Im}(f)=[k]$8, and each vertex $\operatorname{Im}(f)=[k]$9 has one outgoing edge $k\le n$0. Every connected component contains a unique directed cycle, with directed rooted trees feeding into it. A node is internal if it has indegree at least $k\le n$1, equivalently if it lies in the image of $k\le n$2; a node is a leaf if it has indegree $k\le n$3; a fixed point is a loop $k\le n$4 (Cerbai et al., 12 Jul 2025).

In this language, a Cayley permutation is characterized by an initial-segment condition on internal nodes: $k\le n$5 is a Cayley permutation with $k\le n$6 if and only if the internal nodes are exactly $k\le n$7. Hence the leaves are exactly $k\le n$8 (Cerbai et al., 12 Jul 2025). This recasts the packed-word condition as a graph-theoretic labeling constraint: internal nodes must occupy the smallest labels, and all larger labels are leaves.

A fixed-point-free Cayley permutation is therefore a Cayley permutation whose functional digraph has no loops. Equivalently, every connected component still has a unique directed cycle, but every such cycle has length at least $k\le n$9 (Cerbai et al., 12 Jul 2025). This formulation is especially useful because it isolates fixed-point-freeness from the packed-word normalization: the normalization controls which labels are internal, while fixed-point-freeness controls the cycle lengths in the recurrent part.

3. Enumeration by two-sort species

The initial-segment constraint obstructs the naive one-sort decomposition used for permutations and endofunctions. In particular, Cayley permutations are not captured by a direct analogue of $f(1)\cdots f(n)$0, and fixed-point-free Cayley permutations cannot be introduced simply by writing $f(1)\cdots f(n)$1 (Cerbai et al., 12 Jul 2025). The effective solution is a two-sort species formalism that distinguishes internal nodes and leaves.

Let $f(1)\cdots f(n)$2 mark internal nodes and $f(1)\cdots f(n)$3 leaves. The two-sort species $f(1)\cdots f(n)$4 of rooted trees satisfies

$f(1)\cdots f(n)$5

where the correction term $f(1)\cdots f(n)$6 accounts for the fact that a singleton child which is just a root becomes a leaf in the larger tree (Cerbai et al., 12 Jul 2025). For a unisort species $f(1)\cdots f(n)$7, the corresponding $f(1)\cdots f(n)$8-recurrent functional digraphs of two sorts are

$f(1)\cdots f(n)$9

Here the $f(i)\neq i$0-structure sits on the recurrent part, that is, on the roots lying on directed cycles.

The fundamental recursive identity is the partial differential equation

$f(i)\neq i$1

obtained by marking a distinguished leaf and tracing the unique branch that connects it back to the recurrent part (Cerbai et al., 12 Jul 2025). Expanding coefficients gives a recursion for $f(i)\neq i$2, and solving that recursion introduces the difference $f(i)\neq i$3-Stirling numbers $f(i)\neq i$4. The general closed form is

$f(i)\neq i$5

To recover one-sort Cayley permutations, one enforces that internal nodes occupy the initial segment of labels; the resulting $f(i)\neq i$6-recurrent Cayley permutations satisfy

$f(i)\neq i$7

Specializing $f(i)\neq i$8, where $f(i)\neq i$9, yields the main counting formula for fixed-point-free Cayley permutations: $i$0 The first values are

$i$1

for $i$2 (Cerbai et al., 12 Jul 2025).

The same framework produces refined counts for other functional-digraph constraints. Choosing $i$3 counts, respectively, Cayley permutations whose functional digraph is a tree, a forest, connected, or fixed-point-free; in particular,

$i$4

Thus fixed-point-free Cayley permutations arise as one member of a broader $i$5-recurrent family rather than as an isolated subclass (Cerbai et al., 12 Jul 2025).

4. Pattern avoidance, insertion encodings, and other notions of fixed point

Pattern avoidance on Cayley permutations is developed in a separate species framework in which $i$6 denotes the Cayley permutations avoiding a pattern $i$7. That theory provides species equations, generating series, and counting formulas for all patterns of length at most three, but it does not turn fixed-point-freeness into a pattern class, precisely because $i$8 is not a relative-order condition (Claesson et al., 2024). The same paper introduces primitive Cayley permutations, meaning those with no flat steps $i$9, and proves

$[n]$0

This is structurally close to fixed-point questions but not identical to them: flat-step-freeness is local in the word, whereas fixed-point-freeness is positional (Claesson et al., 2024).

Insertion-encoding methods provide a complementary perspective. Vertical and horizontal insertion encodings generalize the Albert–Linton–Ruškuc insertion encoding from permutations to Cayley permutations, classify when the resulting languages are regular, and yield an algorithm for rational generating functions on finitely based classes via slot-boundedness and vertical or horizontal juxtaposition criteria (Bean et al., 13 May 2025). Fixed-point-free Cayley permutations are not treated explicitly there, and the crucial point is again that they do not form a class $[n]$1. Nevertheless, the same framework presents fixed-point-free subfamilies inside regular classes as accessible by adding extra automaton or tiling constraints, rather than by modifying the pattern basis (Bean et al., 13 May 2025).

A third meaning of fixed point appears in the stack-sorting literature. For a $[n]$2-stack operator $[n]$3 and reversal $[n]$4, the map $[n]$5 is an involution precisely when $[n]$6, and then $[n]$7 is bijective on the set $[n]$8 of Cayley permutations (Cerbai, 2020). In that setting, a fixed point is an operator fixed point satisfying

$[n]$9

This operator-theoretic notion is distinct from classical fixed points $[n]$0, and the distinction matters whenever “fixed-point-free Cayley permutation” is read across different subliteratures (Cerbai, 2020).

5. Cycle-parity classes and the Cayley-continuant connection

A different, older use of “Cayley-type” arises from Cayley continuants and parity-restricted cycle classes of ordinary permutations. For permutations on $[n]$1, one considers

$[n]$2

together with the intermediate class

$[n]$3

The even-cycle class $[n]$4 is automatically fixed-point-free, since 1-cycles are odd (Chen, 2023).

The key enumerative identity is

$[n]$5

and the bijective mechanism is the cycle-breaking procedure that splits or merges the cycle containing $[n]$6 and $[n]$7. This yields a bijection $[n]$8, and by iteration a bijection $[n]$9 (Chen, 2023). In this setting, every element of $w_1\cdots w_n$0 is a fixed-point-free Cayley-type permutation in the sense of a Cayley-continuant specialization, but these objects are ordinary permutations with cycle-parity restrictions rather than packed words or surjective endofunctions. The coincidence of terminology therefore reflects adjacent combinatorial origins rather than identity of models.

6. Refined statistics, identities, and open problems

The modern theory of fixed-point-free Cayley permutations is still developing. One open asymptotic question asks whether

$w_1\cdots w_n$1

The analogy is classical: for permutations, $w_1\cdots w_n$2, and for endofunctions the fixed-point-free count is $w_1\cdots w_n$3, so $w_1\cdots w_n$4; the Cayley case lies between these two models, but the limit remains open (Cerbai et al., 12 Jul 2025).

The same two-sort species formalism also leads to identities beyond enumeration. One unisort reformulation is

$w_1\cdots w_n$5

and the specialization $w_1\cdots w_n$6 gives an ordinal-product identity for $w_1\cdots w_n$7. A direct combinatorial proof of this identity is posed as an open problem (Cerbai et al., 12 Jul 2025). Generalized $w_1\cdots w_n$8-models, obtained by replacing linear branches with a species $w_1\cdots w_n$9, are proposed as a further extension capable of handling other endofunction restrictions (Cerbai et al., 12 Jul 2025).

From the pattern-avoidance side, primitive structures provide another route to fixed-point statistics. The species-theoretic conjecture

$f:[n]\to[n]$00

relates primitive Cayley permutations, total letter sums, and the aggregate contribution of fixed-point positions (Claesson et al., 2024). This does not enumerate fixed-point-free Cayley permutations directly, but it exhibits a nontrivial global link between fixed points and the primitive/no-flat-step decomposition.

Taken together, the literature separates the topic into three principal regimes. The endofunction regime gives the direct definition and the current exact counting formula for Cayley-derangements (Cerbai et al., 12 Jul 2025). The species and insertion-encoding regimes show that fixed-point-freeness is not a standard pattern-avoidance condition, but can be studied through marked species, inclusion–exclusion, and automata on regular classes (Claesson et al., 2024, Bean et al., 13 May 2025). The Cayley-continuant regime shows that fixed-point-freeness also appears naturally as the even-cycle extreme of cycle-parity restrictions, with cardinality $f:[n]\to[n]$01 and explicit cycle-breaking bijections (Chen, 2023). The resulting subject is therefore not a single enumerative sequence, but a family of closely related fixed-point-free phenomena organized by the normalization $f:[n]\to[n]$02, the structure of functional digraphs, and the interaction between positional and operator notions of fixed point.