Cayley Permutations: Structure & Applications

• Cayley permutations are words of positive integers whose set of values forms an initial interval, bridging ordinary permutations and arbitrary words.
• They admit bijections with weak orders, ballots, and functional digraphs, enabling species-theoretic decompositions and precise enumerations.
• Cayley permutations support studies in pattern avoidance, descent statistics, and sorting operators, yielding refined combinatorial models and generating functions.

Searching arXiv for recent and foundational papers on Cayley permutations and closely related usages of the term. Cayley permutations are words of positive integers in which the set of values is an initial interval $[k]=\{1,2,\dots,k\}$ for some $k\le n$. Equivalently, for length $n$ they are maps $w:[n]\to[n]$ with image $[k]$, identified with their one-line notation $w_1\cdots w_n$. They interpolate between ordinary permutations and arbitrary words, admit bijective descriptions in terms of ballots and weak orders, and support parallel theories of pattern avoidance, descent statistics, combinatorial species, functional digraphs, and stack-sorting operators (Claesson et al., 2024, Cerbai et al., 2024, Cerbai et al., 12 Jul 2025).

1. Definition and equivalent models

A Cayley permutation of length $n$ is a word $w=w_1\cdots w_n$ such that if a letter $b$ appears, then every smaller positive integer also appears. In the formulation used in several papers, this is equivalent to requiring $\operatorname{Im}(w)=[k]$ for some $k\le n$0. Small examples are

$k\le n$1

and

$k\le n$2

The same class can be viewed in at least three standard ways. First, Cayley permutations are the packed or normalized representatives of order-isomorphism classes of words: one replaces the smallest letter by $k\le n$3, the next smallest by $k\le n$4, and so on. Second, they are in bijection with weak orders on $k\le n$5: the relation $k\le n$6 is total, reflexive, and transitive. Third, they are in bijection with ballots, or ordered set partitions. If

$k\le n$7

is a ballot on $k\le n$8, then the associated Cayley permutation is the map $k\le n$9 defined by $n$0. Conversely, the fibers of a Cayley permutation form a ballot. This gives an isomorphism of species

$n$1

hence the exponential generating function

$n$2

The coefficients are the Fubini numbers, also called ordered Bell numbers (Claesson et al., 2024).

This ballot model is often the most effective structural encoding. A block $n$3 records the positions occupied by the value $n$4, and species operations on ballots translate directly into decomposition formulas for restricted Cayley permutations. That viewpoint underlies much of the modern theory.

2. Species and functional-digraph frameworks

The species-theoretic description $n$5 is only the first layer. A more elaborate framework arises from functional digraphs. In that setting, a Cayley permutation on $n$6 is treated as a function whose internal nodes are exactly the labels $n$7 for some $n$8, while the leaves are $n$9. To express this cleanly, recent work introduces two-sort species $w:[n]\to[n]$0 of $w:[n]\to[n]$1-recurrent functional digraphs,

$w:[n]\to[n]$2

where $w:[n]\to[n]$3 is the species of rooted trees with internal nodes of sort $w:[n]\to[n]$4 and leaves of sort $w:[n]\to[n]$5. Collapsing the two sorts in one way yields endofunctions; collapsing them according to the total order yields Cayley permutations (Cerbai et al., 12 Jul 2025).

This approach gives a uniform treatment of structural subclasses. Choosing the recurrent species $w:[n]\to[n]$6 as the species of permutations, derangements, sets, cycles, or a singleton yields, respectively, all Cayley permutations, fixed-point-free Cayley permutations, forests, connected functional digraphs, and tree-like Cayley permutations. The resulting formulas recover classical endofunction counts on one side and produce genuinely Cayley-permutation enumerations on the other.

A fixed-point-free Cayley permutation is one with no $w:[n]\to[n]$7 such that $w:[n]\to[n]$8. If $w:[n]\to[n]$9 denotes the set of ordinary derangements of size $[k]$0, then the number of fixed-point-free Cayley permutations on $[k]$1 is

$[k]$2

where $[k]$3 is the paper’s difference-of-$[k]$4-Stirling-number statistic. The initial values are

$[k]$5

for $[k]$6 (Cerbai et al., 12 Jul 2025).

The same formalism yields notable specializations. Cayley permutations whose functional digraph is a tree are equinumerous with all Cayley permutations of size $[k]$7,

$[k]$8

For forests and connected functional digraphs, the corresponding counts are given by explicit sums over the same modified $[k]$9-Stirling numbers. This suggests that Cayley permutations behave as an intermediate class between permutations and arbitrary endofunctions, with the functional-digraph decomposition retaining enough rigidity to admit species-level recursion.

3. Pattern avoidance and small-pattern classification

Pattern avoidance for Cayley permutations is defined by order-isomorphic subsequences, with equalities and inequalities both required to match. If $w_1\cdots w_n$0 and $w_1\cdots w_n$1, then $w_1\cdots w_n$2 contains $w_1\cdots w_n$3 if some subsequence $w_1\cdots w_n$4 has the same strict and weak comparison pattern as $w_1\cdots w_n$5; otherwise $w_1\cdots w_n$6 avoids $w_1\cdots w_n$7. Reverse and complement preserve Cayley permutations and generate the basic symmetry classes (Claesson et al., 2024).

A systematic species-based analysis is available for every pattern of length at most three. The classification is especially clean.

Pattern class	Structural description	Enumeration
$w_1\cdots w_n$8	Ordinary permutations	$w_1\cdots w_n$9
$n$0	Weakly monotone blocks / compositions	$n$1 for $n$2
$n$3	$n$4	egf $n$5
$n$6	$n$7 as an $n$8-species	$n$9 for $w=w_1\cdots w_n$0
$w=w_1\cdots w_n$1	One Cayley-equivalence class	ogf $w=w_1\cdots w_n$2

Avoiding $w=w_1\cdots w_n$3 forbids repeated letters, so the class is just $w=w_1\cdots w_n$4. Avoiding $w=w_1\cdots w_n$5 or $w=w_1\cdots w_n$6 forces the word to consist of constant blocks in increasing or decreasing value order, and the resulting count is the number of compositions of $w=w_1\cdots w_n$7, namely $w=w_1\cdots w_n$8. Avoiding $w=w_1\cdots w_n$9 means every value occurs at most twice, which yields the species $b$0.

The six patterns with one repeated letter but not all equal,

$b$1

form a single Cayley-equivalence class. A central result identifies

$b$2

as $b$3-species, so their exponential generating function is

$b$4

and the number of avoiders is $b$5. The proof uses two different decompositions: one through ballots and derivatives, and one through the observation that in a $b$6-avoider all occurrences of the maximal letter form a contiguous block.

The six classical permutation patterns of length three,

$b$7

also form a single Cayley-equivalence class. For these, the paper adapts the Simion–Schmidt bijection by replacing ordinary left-to-right minima with weak left-to-right minima and augmenting them with the multiset of non-minimal entries. The resulting ordinary generating function

$b$8

applies to every pattern in $b$9.

Beyond total counts, several refined equivalence notions are defined: strong-Cayley-equivalence, Cayley-max-equivalence, and Cayley-equivalence. The first two are shown to match the corresponding notions for patterns in words via binomial inversion between $\operatorname{Im}(w)=[k]$0 and $\operatorname{Im}(w)=[k]$1. This suggests that Wilf-type phenomena for Cayley permutations are closely tied to those for packed words, although a full analogue of classical permutation-pattern symmetry remains absent because there is no useful inverse operation on Cayley permutations.

4. Primitive structures and sorting operators

A primitive Cayley permutation is one with no adjacent equal letters. If $\operatorname{Im}(w)=[k]$2 denotes the species of primitive Cayley permutations, then every Cayley permutation is obtained by taking a primitive one and duplicating letters inside constant runs. Species-theoretically,

$\operatorname{Im}(w)=[k]$3

This relation extends to pattern-avoiding subclasses whenever the forbidden pattern is itself primitive. For instance, $\operatorname{Im}(w)=[k]$4, so for each $\operatorname{Im}(w)=[k]$5 there is exactly one primitive $\operatorname{Im}(w)=[k]$6-avoiding Cayley permutation, namely $\operatorname{Im}(w)=[k]$7 (Claesson et al., 2024).

A different but related direction studies stack-sorting and pattern-avoiding machines on Cayley permutations. Given a pattern $\operatorname{Im}(w)=[k]$8, a $\operatorname{Im}(w)=[k]$9-stack is a stack whose top-to-bottom content is required to avoid $k\le n$00, and the associated $k\le n$01-machine is a $k\le n$02-stack followed by a $k\le n$03-stack, operated by a right-greedy rule. If $k\le n$04 is the output of the first stack, then $k\le n$05 is $k\le n$06-sortable exactly when $k\le n$07 avoids $k\le n$08 (Cerbai, 2020).

The sortable class exhibits a sharp dichotomy. If $k\le n$09, obtained from $k\le n$10 by swapping the first two entries, contains $k\le n$11, then

$k\le n$12

so the sortable Cayley permutations form a pattern class with basis either $k\le n$13 or $k\le n$14. The exceptional case is $k\le n$15, where

$k\le n$16

If $k\le n$17 and $k\le n$18 avoids $k\le n$19, then $k\le n$20 is not a class.

The first-stack operator

$k\le n$21

is itself structurally rich. It is bijective if and only if $k\le n$22, and in that case

$k\le n$23

is an involution on the set of Cayley permutations. For $k\le n$24, $k\le n$25 is a length-preserving bijection, $k\le n$26 is an involution, and the number of $k\le n$27-sortable Cayley permutations of length $k\le n$28 equals the number of $k\le n$29-avoiding Cayley permutations of length $k\le n$30.

Two generalized pop-stack models were also analyzed. Hare pop-stack sortable Cayley permutations are exactly those avoiding

$k\le n$31

whereas tortoise pop-stack sortable Cayley permutations are exactly those avoiding

$k\le n$32

In the tortoise case the enumeration is explicit: $k\le n$33 This suggests that allowing repeated values creates new forbidden configurations, such as $k\le n$34, $k\le n$35, and $k\le n$36, that have no analogue for ordinary permutations.

5. Caylerian polynomials and refined enumeration

Caylerian polynomials record descent statistics over Cayley permutations in direct analogy with Eulerian polynomials. For $k\le n$37, define

$k\le n$38

with

$k\le n$39

Then the weak and strict Caylerian polynomials are

$k\le n$40

They satisfy the symmetry

$k\le n$41

which reflects the reverse-complement symmetry on Cayley permutations (Cerbai et al., 2024).

A principal enumerative formula expresses $k\le n$42 in terms of Fubini numbers and Stirling numbers of the second kind: $k\le n$43 The strict version is the alternating analogue

$k\le n$44

The same paper derives Carlitz-type generating-function identities. If $k\le n$45 and $k\le n$46 denote the appropriate Burge-matrix classes, then

$k\le n$47

After summing over $k\le n$48, this becomes

$k\le n$49

and

$k\le n$50

These formulas are obtained through a species theory of Burge matrices, matrices of linear orders, and sign-reversing involutions. Two-sided refinements

$k\le n$51

simultaneously track nonzero rows and columns, and specialize back to Caylerian polynomials through

$k\le n$52

A plausible implication is that the descent theory of Cayley permutations is most naturally linearized not by direct word arguments but by matrix-valued species.

6. Terminology and adjacent graph-theoretic usages

In contemporary combinatorics, “Cayley permutation” usually denotes the packed-word object described above. In several graph-theoretic and coding-theoretic papers, however, the phrase is used more loosely for permutations regarded as vertices of Cayley graphs or as words in $k\le n$53 equipped with the Cayley metric. Those usages are distinct.

In permutation coding theory, the Cayley distance between $k\le n$54 is the minimum number of transpositions needed to transform $k\le n$55 into $k\le n$56, with the formula

$k\le n$57

A 2024 paper improves the Gilbert–Varshamov lower bound for permutation codes in the Cayley metric and in the Kendall $k\le n$58-metric by a factor of $k\le n$59, obtaining

$k\le n$60

for fixed $k\le n$61 (Nguyen, 2024).

In the theory of Cayley graphs generated by transpositions, permutations are the vertices of $k\le n$62 with $k\le n$63. When the transposition graph $k\le n$64 has girth at least $k\le n$65, the full automorphism group satisfies

$k\le n$66

and, when the connected components of $k\le n$67 are isomorphic, $k\le n$68 (Ganesan, 2013).

A related literature studies Cayley graphs of $k\le n$69 generated by transposition trees. There the vertices are again permutations, the graph distance is the minimum number of allowed tree-transpositions, and the diameter is bounded by

$k\le n$70

with additional polynomial-time diameter estimates based only on the tree (Ganesan, 2011).

These graph-theoretic usages do not redefine the packed-word notion. They instead place ordinary permutations inside Cayley graphs, Cayley metrics, or Cayley-generated networks. The coexistence of both vocabularies is historically understandable, but the underlying objects are different: one concerns surjective packed words and their species, the other concerns the symmetric group as a metric or graph-theoretic object.