Fishburn Permutations in Combinatorics

• Fishburn permutations are defined by avoiding the bivincular pattern (231, {1}, {1}) and are enumerated by the Fishburn numbers.
• They are bijectively linked to ascent sequences, Fishburn matrices, and (2+2)-free posets, facilitating cross-structural analyses.
• Pattern avoidance in this class yields refined enumerations, revealing Catalan, Fibonacci, and Pell subclasses that inspire ongoing research.

Fishburn permutations are permutations avoiding the bivincular pattern $(231,\{1\},\{1\})$. They constitute one of the central families counted by the Fishburn numbers, together with ascent sequences, $(2+2)$-free posets, Fishburn matrices, and several matching classes. The subject lies at the junction of bivincular pattern avoidance, bijective combinatorics, and refined enumeration, and the modern literature treats Fishburn permutations both as a permutation class in its own right and as one node in a dense web of bijections linking several canonical Fishburn structures (Gil et al., 2018, Levande, 2010).

1. Definition and basic enumerative framework

A Fishburn permutation is a permutation that avoids the bivincular pattern $(231,\{1\},\{1\})$. One explicit formulation states that a permutation $\pi=\pi_1\pi_2\ldots\pi_n$ contains the Fishburn pattern if there are indices $j+1<k$ such that $\pi_j=\pi_k+1$, $\pi_{j+1}>\pi_j$, and $(\pi_j,\pi_{j+1},\pi_k)$ is order isomorphic to $231$; Fishburn permutations are those for which no such configuration exists (Egge, 2022). An equivalent formulation used elsewhere is that there do not exist $i<j-1$ such that $(2+2)$0 (Lv et al., 11 Sep 2025).

The class is counted by the Fishburn numbers. In the notation of Gil and Weiner, if $(2+2)$1 denotes the set of Fishburn permutations of length $(2+2)$2, then $(2+2)$3, where the generating function of the Fishburn numbers is

$(2+2)$4

The same sequence also enumerates ascent sequences, chord diagrams, and $(2+2)$5-free posets, which is the combinatorial reason Fishburn permutations are usually studied inside a broader “Fishburn world” rather than in isolation (Gil et al., 2018).

The literature uses both $(2+2)$6 and $(2+2)$7 for subclasses avoiding an additional classical pattern $(2+2)$8. This dual notation reflects the fact that the theory developed partly from permutation-pattern questions and partly from bijections with ascent sequences and interval orders.

2. Bijections with ascent sequences and related encodings

The foundational structural fact is the bijection between Fishburn permutations and ascent sequences. Bousquet-Mélou, Claesson, Dukes, and Kitaev constructed a natural bijection $(2+2)$9 between Fishburn permutations of length $(231,\{1\},\{1\})$0 and ascent sequences of length $(231,\{1\},\{1\})$1; in this construction the permutation is built by successively inserting $(231,\{1\},\{1\})$2 into active sites, and $(231,\{1\},\{1\})$3 records the labels of these insertions (Egge, 2022). This active-site description is one of the standard mechanisms for translating permutation avoidance into sequence avoidance.

A more explicit factorization of the correspondence proceeds through Fishburn matrices. In the bijection of Chen, Yan, and Zhou, one first maps a Fishburn permutation $(231,\{1\},\{1\})$4 to an ascent sequence $(231,\{1\},\{1\})$5, where $(231,\{1\},\{1\})$6 is the active-site label of the insertion of $(231,\{1\},\{1\})$7; one then maps ascent sequences to Fishburn matrices by a recursive map $(231,\{1\},\{1\})$8 using three matrix operations, Add1, Add2, and Add3; and finally applies a flip $(231,\{1\},\{1\})$9 along the North-East diagonal. The composite map is

$\pi=\pi_1\pi_2\ldots\pi_n$0

This produces an explicit bijection from Fishburn permutations to Fishburn matrices (Chen et al., 2018).

Modified ascent sequences provide another useful encoding. The Burge transpose gives a mechanism for transporting pattern-avoidance questions from Fishburn permutations to modified ascent sequences and, more generally, to Cayley permutations. For every permutation pattern $\pi=\pi_1\pi_2\ldots\pi_n$1, the transport theorem gives

$\pi=\pi_1\pi_2\ldots\pi_n$2

where $\pi=\pi_1\pi_2\ldots\pi_n$3 is an explicitly constructed Fishburn basis in the modified ascent-sequence setting (Cerbai et al., 2020). This transport machinery reframes many Fishburn-pattern problems as basis-computation problems on ascent-sequence-like objects.

Fishburn trees supply a more recent hub object. A Fishburn tree is a regular endotree $\pi=\pi_1\pi_2\ldots\pi_n$4 satisfying $\pi=\pi_1\pi_2\ldots\pi_n$5, and its in-order traversal yields a modified ascent sequence. The paper on Fishburn trees gives explicit bijections between Fishburn trees, modified ascent sequences, Fishburn matrices, and $\pi=\pi_1\pi_2\ldots\pi_n$6-free posets; the relation to Fishburn permutations is indirect and goes via the Burge transpose on modified ascent sequences (Cerbai et al., 2022). This suggests a geometric reorganization of the classical Fishburn correspondences around tree structure.

3. Fishburn permutations among Fishburn objects

Because Fishburn permutations sit inside a network of bijections, many permutation statistics admit matrix, poset, or matching interpretations. Under the permutation–matrix bijection of Chen, Yan, and Zhou, the four-tuple

$\pi=\pi_1\pi_2\ldots\pi_n$7

on Fishburn permutations corresponds exactly to

$\pi=\pi_1\pi_2\ldots\pi_n$8

on Fishburn matrices. Thus the number of left-to-right maxima becomes the weight of the first row, the number of right-to-left minima becomes the last-column weight, the number of right-to-left maxima becomes the number of weakly North-East cells, and the number of left-to-right minima becomes the number of nonzero diagonal cells (Chen et al., 2018).

This translation is part of a wider theory of equidistribution across Fishburn structures. Fishburn triples were introduced as relational structures $\pi=\pi_1\pi_2\ldots\pi_n$9 that generalize two types of Catalan pairs and encode interval orders and Fishburn matrices. In that framework, several Catalan equidistribution phenomena extend to Fishburn statistics such as wNE, sNE, wSE, and sSE cells. For Fishburn permutations the relevant bijection was still conjectural in that paper, which is historically notable because later work supplied explicit permutation–matrix correspondences (Jelínek, 2015).

Enumeration at the level of the Fishburn distribution is also not confined to permutations. A sieve-based method constructs Fishburn-distributed statistics from Mahonian structures and yields the identity

$j+1<k$0

expressing Fishburn numbers in terms of Mahonian numbers. The same paper introduces zero alignment matchings and mislabelings in factorial posets as further carriers of the Fishburn distribution (Hannah, 2015). In this sense, Fishburn permutations are one manifestation of a much broader enumerative schema.

Asymptotic results for Fishburn matrices transfer to Fishburn permutations through these bijections. One paper emphasizes that any enumerative or statistical result for Fishburn matrices automatically gives a corresponding result for Fishburn permutations (Hwang et al., 2019). A later paper makes this more concrete by stating that the dimension of a Fishburn matrix is equidistributed with statistics including the number of descents in the associated Fishburn permutation; the asymptotic normality established for matrix dimension therefore has a direct permutation interpretation (Hwang et al., 2020).

4. Classical pattern avoidance and Wilf classes

Pattern avoidance inside the Fishburn class has a distinctive enumerative profile. For classical patterns of size $j+1<k$1, Gil and Weiner give a complete picture. If $j+1<k$2, then

$j+1<k$3

If $j+1<k$4, then

$j+1<k$5

where $j+1<k$6 is the Catalan number. If $j+1<k$7, then

$j+1<k$8

The same paper also treats indecomposable Fishburn permutations and derives, for example,

$j+1<k$9

For $\pi_j=\pi_k+1$0, the indecomposable sequence satisfies $\pi_j=\pi_k+1$1 with $\pi_j=\pi_k+1$2 (Gil et al., 2018).

For size $\pi_j=\pi_k+1$3 patterns, one of the most important phenomena is a Catalan Wilf class. The patterns

$\pi_j=\pi_k+1$4

are all Wilf-equivalent within Fishburn permutations, each with enumeration $\pi_j=\pi_k+1$5. By contrast, the pattern $\pi_j=\pi_k+1$6 yields the binomial transform of the Catalan numbers: $\pi_j=\pi_k+1$7 The same paper also records conjectural Wilf equivalences such as

$\pi_j=\pi_k+1$8

These results show that the Fishburn restriction substantially reorganizes classical Wilf-equivalence behavior rather than merely inheriting it (Gil et al., 2018).

Avoided pattern(s)	Enumeration
$\pi_j=\pi_k+1$9	$\pi_{j+1}>\pi_j$0
$\pi_{j+1}>\pi_j$1	$\pi_{j+1}>\pi_j$2
$\pi_{j+1}>\pi_j$3	explicit summation formula above
$\pi_{j+1}>\pi_j$4	$\pi_{j+1}>\pi_j$5
$\pi_{j+1}>\pi_j$6	$\pi_{j+1}>\pi_j$7

5. Refined subclasses, generating trees, and recent enumerations

Later work sharpened the Fishburn pattern-avoidance theory in several directions. One result settles a conjecture of Gil and Weiner by proving that the bijection $\pi_{j+1}>\pi_j$8 restricts to

$\pi_{j+1}>\pi_j$9

The same paper shows that every permutation in $(\pi_j,\pi_{j+1},\pi_k)$0 corresponds, under $(\pi_j,\pi_{j+1},\pi_k)$1, to a binary ascent sequence, giving $(\pi_j,\pi_{j+1},\pi_k)$2, and then refines the enumeration by inversion number and number of left-to-right maxima using $(\pi_j,\pi_{j+1},\pi_k)$3-binomial coefficients. Generating tree methods further yield

$(\pi_j,\pi_{j+1},\pi_k)$4

$(\pi_j,\pi_{j+1},\pi_k)$5

and

$(\pi_j,\pi_{j+1},\pi_k)$6

The paper also gives formulas for $(\pi_j,\pi_{j+1},\pi_k)$7 across all $(\pi_j,\pi_{j+1},\pi_k)$8 (Egge, 2022).

A separate paper proves two conjectures of Egge concerning classes that avoid $(\pi_j,\pi_{j+1},\pi_k)$9 together with additional patterns. Among the exact formulas obtained are

$231$0

$231$1

$231$2

$231$3

$231$4

$231$5

and, for $231$6,

$231$7

where $231$8 is Fibonacci and $231$9 is Pell (Du et al., 2023).

The Catalan subclass $i<j-1$0 has become especially prominent. It is counted by the Catalan numbers and is linked, via explicit bijections, to Stoimenow matchings avoiding $i<j-1$1, $i<j-1$2-free posets, and $i<j-1$3-avoiding ascent sequences. Over this subclass, the statistic $i<j-1$4 satisfies

$i<j-1$5

so right-to-left minima are distributed by the Narayana numbers. The same paper gives a symmetric joint distribution involving $i<j-1$6 and $i<j-1$7, and the generating function

$i<j-1$8

where $i<j-1$9 is the generating function for ballot numbers. It also records a Catalan decomposition: if $(2+2)$00, then

$(2+2)$01

with $(2+2)$02 and $(2+2)$03 again in the $(2+2)$04-avoiding Fishburn class (Lv et al., 11 Sep 2025).

6. Extensions, transports, and open directions

One recent extension generalizes ascent sequences to $(2+2)$05-ascent sequences and defines modified $(2+2)$06-ascent sequences through a $(2+2)$07-hat map. Composed with the Burge transpose, this yields a size-preserving bijection

$(2+2)$08

from $(2+2)$09-ascent sequences to $(2+2)$10-Fishburn permutations. In the classical case $(2+2)$11, the paper proves that Fishburn permutations are exactly the ir-subdiagonal permutations: if a permutation is decomposed into maximal increasing runs $(2+2)$12, then every letter $(2+2)$13 must satisfy $(2+2)$14. This gives a new characterization of Fishburn permutations that is not phrased in terms of bivincular avoidance (Cerbai et al., 2024).

Pattern transport remains an active structural theme. The Burge-transpose framework provides an explicit basis-construction procedure sending a permutation pattern $(2+2)$15 to a basis $(2+2)$16 for modified ascent sequences. This does not merely transfer enumerations; it supplies a systematic mechanism for translating avoidance questions between Fishburn permutations and Cayley-permutation models (Cerbai et al., 2020).

Fishburn permutations also appear indirectly in arithmetic generalizations of Fishburn numbers. Generalized Fishburn numbers $(2+2)$17 are defined as the coefficients in the $(2+2)$18 expansion of the Kontsevich–Zagier series $(2+2)$19 attached to torus knots $(2+2)$20, and the paper establishing prime power congruences for these numbers notes that classical Fishburn numbers count Fishburn permutations and interval orders. At the same time, it emphasizes that combinatorial interpretations for the generalized numbers are not yet established and identifies the search for such interpretations as an open direction (Bijaoui et al., 2020).

Taken together, these developments position Fishburn permutations as both a concrete bivincular-avoidance class and a transport object for a larger theory. The class supports exact enumeration, refined statistics, bijections with matrices, posets, trees, and ascent-sequence models, Catalan/Fibonacci/Pell subfamilies, and generalized variants such as $(2+2)$21-Fishburn permutations. The persistent open problems concern direct combinatorial models for new Fishburn-type sequences, further Wilf classifications, and deeper statistic-preserving bijections across the Fishburn correspondence network.