Stoimenow Matchings: Fishburn Connections

• Stoimenow matchings are perfect matchings on [2n] that avoid neighbor nestings, serving as key objects in enumerative combinatorics.
• They are in bijection with (2+2)-free posets, ascent sequences, Fishburn permutations, and matrices, leading to the Fishburn number sequence.
• One-sided relaxations and pattern avoidances reveal factorial, Catalan, and Fibonacci regimes, underpinning rich asymptotic and statistical analyses.

Searching arXiv for recent and foundational papers on Stoimenow matchings. Stoimenow matchings are perfect matchings on the ordered set $[2n]=\{1,2,\dots,2n\}$, drawn as arcs above a line, that avoid neighbor nestings: neither a left-nesting nor a right-nesting is permitted. They were introduced by Stoimenow under the name regular linearized chord diagrams in connection with bounds for Vassiliev knot invariants, and they have become a standard family of Fishburn objects, in bijection with $(2+2)$-free posets, ascent sequences, Fishburn permutations, and Fishburn matrices. Their enumeration is given by the Fishburn numbers, while several natural subclasses and one-sided relaxations exhibit factorial, Catalan, Fibonacci, and other enumerative regimes (Hwang et al., 2020, Lv et al., 11 Sep 2025).

1. Definition and local structure

A matching on $[2n]$ is a partition into $n$ blocks of size $2$. Writing an arc as $(i,j)$ with $i<j$, the point $i$ is the opener and $j$ the closer. Two arcs $(i,\ell)$ and $(2+2)$0 form a nesting if $(2+2)$1, and they form a crossing if $(2+2)$2 when written instead as $(2+2)$3 and $(2+2)$4 (Claesson et al., 2010).

The neighbor-restricted versions are the defining local obstructions. A left-nesting is a nesting with $(2+2)$5, so the openers are consecutive; a right-nesting is a nesting with $(2+2)$6, so the closers are consecutive. Stoimenow matchings are precisely those matchings with no neighbor nestings, meaning no left-nesting and no right-nesting. In the terminology of later work, these are also the matchings avoiding Type I and Type II local configurations (Claesson et al., 2010, Lv et al., 11 Sep 2025).

This local definition is weaker than global nonnesting. Stoimenow matchings may contain arbitrary crossings and may also contain nestings, provided that the nested pair does not have adjacent openers or adjacent closers. A common misconception is that they coincide with nonnesting matchings; they do not. For example, the fully nested matching $(2+2)$7 is excluded because it contains neighbor nestings, but many other nested configurations remain admissible (Hwang et al., 2020).

The local prohibition has a strong structural flavor. It constrains how arcs can be inserted or deleted near adjacent endpoints, which is why Stoimenow matchings sit naturally inside recursive constructions and bijections with other Fishburn families. This suggests that their combinatorics is governed less by global planarity than by finely controlled local adjacency patterns.

2. Fishburn enumeration and the web of equivalent structures

Stoimenow matchings are one of the canonical classes counted by the Fishburn numbers. Their ordinary generating function is

$(2+2)$8

a form first derived by Zagier for Stoimenow matchings and later situated inside a broader bijective theory (Claesson et al., 2010, Lv et al., 11 Sep 2025).

The central equivalences place Stoimenow matchings in bijection with four other families:

Structure	Description
$(2+2)$9-free posets	Interval orders
Ascent sequences	Sequences $[2n]$0 with $[2n]$1 and $[2n]$2
Fishburn permutations	Permutations avoiding a specific bivincular pattern
Fishburn matrices	Upper-triangular matrices with no zero row or column

These correspondences were established in the Fishburn literature summarized by Claesson–Linusson and later asymptotic work, and they allow statistics, generating functions, and structural decompositions to be transported across very different-looking models (Claesson et al., 2010, Hwang et al., 2020).

For permutations, the relevant forbidden pattern $[2n]$3 is the existence of indices $[2n]$4 with $[2n]$5, $[2n]$6, and $[2n]$7. For ascent sequences, the defining growth restriction is recursive, but the bijections refine the Fishburn generating function by tracking statistics such as ascents. On the poset side, the equivalent objects are unlabeled $[2n]$8-free posets, or interval orders, characterized by the fact that their predecessor sets are linearly ordered by inclusion (Claesson et al., 2010).

The significance of this web of equivalences is methodological as much as enumerative. A matching formulation is especially effective for local endpoint restrictions, a poset formulation for order-theoretic characterizations, and a matrix formulation for analytic and asymptotic arguments. Much of the subsequent literature exploits precisely this change of viewpoint.

3. One-sided relaxations: left-nesting-free matchings and factorial posets

A decisive extension of the Stoimenow framework is the study of matchings that forbid only left-nestings. Let

$[2n]$9

Claesson and Linusson proved that these matchings are in bijection with inversion tables of length $n$0, that is,

$n$1

Consequently,

$n$2

This is Theorem 1 of "n! matchings, n! posets" (Claesson et al., 2010).

The bijection is constructive. Given an inversion table $n$3, one adds a new last arc whose closer is $n$4 and places its opener immediately to the left of the $n$5-st closer already present, except in the case $n$6, where the opener is placed immediately to the left of its own closer. Ordering arcs by increasing closer yields the inverse map: the $n$7-th entry counts the number of closers to the left of the opener of the $n$8-th arc (Claesson et al., 2010).

The corresponding poset class is that of factorial posets. A naturally labeled poset $n$9 on $2$0 is factorial if

$2$1

Equivalently, for every $2$2, the predecessor set has the form

$2$3

for some $2$4. These posets are $2$5-free, and they are again in bijection with inversion tables, hence also counted by $2$6 (Claesson et al., 2010).

This one-sided theory clarifies an important distinction. Forbidding both neighbor nestings yields Fishburn numbers; forbidding only left-nestings yields factorial growth. The comparison is already visible for $2$7: there are $2$8 left-nesting-free matchings on $2$9, namely $(i,j)$0 and $(i,j)$1, whereas the nested matching $(i,j)$2 is excluded by the left-nesting condition (Claesson et al., 2010). A plausible implication is that the right-neighbor condition is the source of the specifically Fishburn, rather than factorial, complexity.

Claesson–Linusson also identified a natural subset of factorial posets satisfying

$(i,j)$3

and proved that this subclass is in bijection with unlabeled $(i,j)$4-free posets. Thus the Fishburn class appears inside the factorial theory as a canonical labeled subfamily (Claesson et al., 2010).

4. Crossings, matrix encodings, and local dualities

The same paper established several bijections showing that neighbor restrictions on nestings can be translated into corresponding restrictions on crossings. In particular, matchings on $(i,j)$5 with no left-crossing are also in bijection with inversion tables $(i,j)$6, so they too are counted by $(i,j)$7 (Claesson et al., 2010).

A more global encoding uses upper-triangular integer matrices. For a matching $(i,j)$8, partition the openers and closers into maximal intervals

$(i,j)$9

and define a matrix $i<j$0 by letting $i<j$1 be the number of arcs whose opener lies in $i<j$2 and whose closer lies in $i<j$3. If $i<j$4 denotes the set of upper-triangular matrices with nonnegative entries, no zero row or column, and total sum $i<j$5, then $i<j$6 is a bijection from matchings with no neighbor nestings onto $i<j$7, and also a bijection from matchings with no neighbor crossings onto $i<j$8 (Claesson et al., 2010).

The block behavior is forced by a key lemma: within a fixed opener interval, no-left-nesting implies that arcs must cross pairwise, while no-left-crossing implies that they must nest pairwise; dually, analogous statements hold for closer intervals under right restrictions. This yields invertibility of the matrix map on the restricted classes (Claesson et al., 2010).

A zero-one subclass $i<j$9 also appears naturally. Matchings with no left-nestings and no right-crossings correspond bijectively to $i$0, linking mixed local restrictions to a particularly rigid matrix family (Claesson et al., 2010).

The relation between nesting and crossing restrictions also admits a uniform explanation via Sundaram’s bijection from matchings to oscillating tableaux. Conjugation of partitions induces an involution on matchings that interchanges right-nestings with right-crossings, and likewise left-nestings with left-crossings. This places the three levels of restriction—full, one-sided, and mixed—inside a single symmetric framework (Claesson et al., 2010).

The matrix perspective became especially influential because Fishburn matrices later supported refined asymptotic analysis. It also clarifies why Stoimenow matchings belong to a broader family of upper-triangular incidence structures, rather than constituting an isolated chord-diagram phenomenon.

5. Catalan subclasses and pattern avoidance

Recent work shifted attention from the defining neighbor-nesting condition to additional forbidden submatchings inside Stoimenow matchings. Lv, Kitaev, and Zhang exhibited five specific $i$1-arc patterns $i$2 such that avoidance of any one of them yields a Catalan class: $i$3 where

$i$4

For $i$5, avoidance coincides with nonnesting matchings; for $i$6, the proof uses a gluing/splitting decomposition that realizes the Catalan recurrence; and for $i$7–$i$8, explicit Wilf-equivalences complete the argument (Lv et al., 11 Sep 2025).

The same paper introduced four infinite families $i$9 generalizing four of those Catalan patterns and proved

$j$0

These equivalences are realized by moving certain consecutive openers within maximal crossing blocks, showing that local pattern movement can preserve global Stoimenow admissibility (Lv et al., 11 Sep 2025).

The Catalan subclasses transfer across the Fishburn correspondences. Under the bijection $j$1 from Stoimenow matchings to $j$2-free posets, $j$3-avoiders correspond to $j$4-free posets, and $j$5-avoiders correspond to $j$6-free posets. Via the classical map from posets to ascent sequences, $j$7-avoiding Stoimenow matchings correspond to ascent sequences avoiding $j$8, and via modified ascent sequences and Burge transpose they correspond to Fishburn permutations avoiding $j$9 (Lv et al., 11 Sep 2025).

A later paper completed the simultaneous-avoidance analysis for subsets of $(i,\ell)$0. The resulting ordinary generating functions fall into nine OEIS sequences. Among the pair classes, for example,

$(i,\ell)$1

so the coefficients are $(i,\ell)$2 for $(i,\ell)$3; the full quintuple-avoidance class has generating function

$(i,\ell)$4

hence eventually quadratic growth (Lv et al., 16 Sep 2025).

These results show that Stoimenow matchings support a surprisingly rich pattern-avoidance theory internal to the Fishburn world. This suggests that Catalan behavior in the class is not confined to global nonnesting, but can also arise from specific local $(i,\ell)$5-arc exclusions.

6. Statistics, asymptotics, and current directions

Fishburn-matrix methods have supplied the first asymptotic limit laws for Stoimenow-type statistics. Let $(i,\ell)$6 be the number of Stoimenow matchings with $(i,\ell)$7 arcs. For ordinary Fishburn matrices, and hence for regular linearized chord diagrams, the asymptotic form is

$(i,\ell)$8

This follows from a saddle-point analysis built on positivity-preserving $(i,\ell)$9-series transformations due to Andrews and Jelínek (Hwang et al., 2020).

The same paper proved a central limit theorem for Fishburn-matrix dimension at fixed size. Through the standard equidistribution table, this transfers to the Stoimenow statistic “length of the initial run of openers.” If $(2+2)$00 denotes that statistic for a uniformly random Stoimenow matching of size $(2+2)$01, then

$(2+2)$02

The same analysis resolved Stoimenow’s conjecture on connected regular linearized chord diagrams: if $(2+2)$03 is the number of connected Stoimenow matchings of size $(2+2)$04, then

$(2+2)$05

Thus a random large Stoimenow matching is connected with asymptotic probability $(2+2)$06 (Hwang et al., 2020).

On the bijective side, Claesson–Linusson tracked several statistics across factorial posets, inversion tables, permutations, and left-nesting-free matchings. They showed, among other correspondences, that components, minima, and several Mahonian and Eulerian parameters are preserved: $(2+2)$07 on factorial posets corresponds to $(2+2)$08 on permutations and $(2+2)$09 on left-nesting-free matchings; $(2+2)$10, $(2+2)$11, and $(2+2)$12 are Eulerianly distributed (Claesson et al., 2010).

More recent Catalan-subclass work added Narayana and ballot refinements. On the classes $(2+2)$13 and $(2+2)$14, statistics such as the number of maximal crossings, largest noncrossing size, and the number of irreducible blocks admit refined generating functions matching Narayana polynomials and ballot-number factorizations, and these distributions transport to corresponding statistics on posets, ascent sequences, and Fishburn permutations (Lv et al., 11 Sep 2025).

Several problems remain open. Claesson–Linusson conjectured equidistributions linking right-nestings on left-nesting-free matchings with the bivincular pattern $(2+2)$15 on permutations and with violations of the canonical-labeling condition on factorial posets; these were checked computationally for $(2+2)$16 (Claesson et al., 2010). In the ascent-sequence setting, the pattern sets $(2+2)$17, $(2+2)$18, and $(2+2)$19 remain unresolved, and the corresponding Stoimenow matching subclasses are therefore also not yet enumerated (Liu et al., 8 Apr 2026). Taken together, these directions indicate that the modern theory of Stoimenow matchings lies at the intersection of local forbidden-configuration theory, Fishburn bijections, and analytic combinatorics, with substantial room for further unification.