Noncrossing Partial Matchings

• Noncrossing partial matchings are set partitions of a finite ordered set into singletons or paired arcs drawn above a line so that no two arcs cross.
• Their enumeration uses binomial coefficients and Catalan numbers, revealing connections to lattice paths like Motzkin paths and refined Catalan theory.
• They exhibit structural bijections with Richardson tableaux and have applications in geometric graph theory, including NP-completeness and extremal results.

A noncrossing partial matching is a set partition of a finite linearly ordered set (usually $[n]=\{1,2,\dots,n\}$) into blocks of size 1 (singletons) or 2 (arcs), such that in the diagrammatic representation—labeling vertices $1,\dots,n$ in sequence and drawing each block of size 2 as a semicircular arc above the line—no two arcs cross. This structure appears naturally across enumerative combinatorics, geometric graph theory, and algebraic combinatorics, forming a cornerstone for connections with lattice path enumeration, standard tableaux, Motzkin paths, and refined Catalan theory (Guo, 19 Nov 2025, Chen et al., 2010, Pilz et al., 2020).

1. Formal Definitions and Diagrammatic Criteria

A partial matching on $[n]$ is a set partition of $[n]$ into singletons and pairs. Each such object can be canonically encoded by a fixed-point-free involution of $S_n$ (permutations $w$ with $w^2 = \mathrm{id}$), where

$$w(i) = \begin{cases} i & \text{if } i \text{ is a singleton} \ j & \text{if } \{i, j\} \text{ is a pair} \end{cases}$$

(Guo, 19 Nov 2025). For each pair $(i, j)$, an arc is drawn above the underlying line; singletons are depicted as isolated points.

A crossing occurs if there are distinct arcs $(i_1, j_1)$, $(i_2, j_2)$ with $i_1 < i_2 < j_1 < j_2$; a noncrossing partial matching is one in which no such configuration occurs, equivalently, whose arc diagram admits no intersecting arcs. This definition restricts the class of partial matchings to those whose arc diagrams are planar—no two edges overlap except possibly at endpoints (Guo, 19 Nov 2025, Chen et al., 2010).

2. Enumerative and Combinatorial Properties

Noncrossing partial matchings generalize noncrossing perfect matchings (which correspond to the classical Catalan numbers) by allowing singletons. The enumeration is explicitly given by

$$|\mathrm{NC}(n)| = \sum_{\substack{0 \le k \le n\ n-k \equiv 0 \pmod 2}} \binom{n}{k} C_{(n-k)/2}$$

where $k$ is the number of singletons and $C_m = \frac{1}{m+1} \binom{2m}{m}$ is the $m$th Catalan number (Guo, 19 Nov 2025). The ordinary generating function for $|\mathrm{NC}(n)|$ is

$$\sum_{n\ge 0} |\mathrm{NC}(n)| x^n = \frac{1-x - \sqrt{1-2x-3x^2}}{2x^2}$$

providing a closed analytic description (Guo, 19 Nov 2025).

A refined enumeration arises in the context of 12312-avoiding partial matchings, where the number of noncrossing matchings with $j$ arcs among $i$ points is

$L_{i, j}(0)$

using the lattice polynomial $L_{i, j}(x)$, which counts certain restricted lattice paths from $(0, 0)$ to $(i, j)$ that never cross above the line $x=2y$ (Chen et al., 2010).

3. Structural Bijections: Motzkin Paths and Richardson Tableaux

There exists a natural bijection between noncrossing partial matchings on $[n]$ and Motzkin paths of length $n$. For each vertex $i$, an up-step ($1,1$) is assigned if $i$ is the left endpoint of an arc, a down-step ($1,-1$) if it is the right endpoint, and a horizontal step ($1,0$) for singletons. This construction yields a Motzkin path, capitalizing on the noncrossing property to ensure nonnegativity and endpoint return (Guo, 19 Nov 2025).

A further, highly structured connection is provided by the Robinson–Schensted (RS) insertion. The insertion tableau of a noncrossing involution $w$ (corresponding to a noncrossing partial matching) is a Richardson tableau— a distinguished subfamily of standard Young tableaux characterized by a strong row-maximality property in their construction (Guo, 19 Nov 2025). The induced map

$$\Phi: \mathrm{NC}(n) \longrightarrow \mathrm{RT}(n), \quad \Phi(w) = \mathrm{Ins}(w)$$

is a bijection, and, when combined with the Motzkin path bijection, gives a natural correspondence between Motzkin paths and Richardson tableaux of size $n$. This settles the open problem of Karp and Precup on indexing Richardson tableaux by Motzkin paths (Guo, 19 Nov 2025).

4. Extremal and Algorithmic Results in Geometric Contexts

Noncrossing partial matchings have equivalents in geometric graph theory as maximal sets of interior-disjoint, noncrossing chords in convex polygons or more general geometric graphs. Important complexity and extremal results include:

• NP-completeness: Deciding whether there exists a compatible perfect matching (i.e., a noncrossing perfect matching) inside a given simple polygon is NP-complete, as is deciding whether a geometric graph can be augmented to minimum degree five by compatible, noncrossing edges (Pilz et al., 2020).
• Extremal sizes: In any $n$-vertex simple polygon, no maximal compatible matching can have fewer than $n/7$ edges; there exist infinite families where this lower bound is attained exactly. For $d$-regular geometric graphs ($d=0,1,2$), minimal maximal compatible matching sizes are bounded below by $\frac{n-1}{3}$, $\frac{n-2}{6}$, and $\frac{n-3}{11}$, respectively (Pilz et al., 2020).

Geometric Graph Type	Minimum Size of Maximal Compatible Matching
Polygon	$\lceil n/7 \rceil$
$d=0$ (empty)	$\lceil (n-1)/3 \rceil$
$d=1$ (perfect)	$\lceil (n-2)/6 \rceil$
$d=2$ (cycle union)	$\lceil (n-3)/11 \rceil$

These results establish tight bounds and serve as geometric analogues to enumerative phenomena encountered in purely combinatorial settings.

5. Pattern Avoidance, Lattice Polynomials, and Further Restrictions

Noncrossing partial matchings can be interpreted as 12312-avoiding partial matchings in the canonical sequence representation—a constraint that precludes the occurrence of subsequences order-isomorphic to $1,2,3,1,2$ in arc-label scans (Chen et al., 2010). The lattice polynomial framework encodes finer distributional data, such as the number of matchings with a given number of arcs and crossings. Specializing the crossing variable $x=0$ recovers the noncrossing case as a subclass of interest.

These restriction classes are in bijection with specific subclasses of lattice paths and trees, and they admit further combinatorial statistics, e.g., “r-index” on even trees corresponding to arc configurations in matchings (Chen et al., 2010).

6. Connections to Tableaux Theory and $q$-Analogues

The bijection between noncrossing partial matchings and Richardson tableaux yields enumerative consequences, such as a formula expressing the number of Richardson tableaux with $k$ odd columns: $\#\{\,T\in\mathrm{RT}(n): \text{%%%%48%%%% has %%%%49%%%% odd columns}\} = \binom{n}{k} C_{(n-k)/2}, \quad k \equiv n \pmod{2}$ (Guo, 19 Nov 2025). For even matchings with no fixed points, the distribution of the comajor index over the corresponding tableaux is precisely the $q$-Catalan number: $\sum_{T\in\mathrm{ERT}(2n)} q^{\comaj(T)} = C_n(q) = \frac{1}{[n+1]_q} \binom{2n}{n}_q$ with further refinements providing $q$-Narayana numbers. There is an explicit conjectural $q$-analogue for arbitrary numbers of fixed points: $\sum_{T\in\mathrm{RT}(n),\;\#\mathrm{odd}(T)=k} q^{\comaj(T)} = q^{\binom{k}{2}}\binom{n}{k}_q C_{(n-k)/2}(q)$ (Guo, 19 Nov 2025).

Additionally, the set of Richardson tableaux is closed under Schützenberger evacuation, and the number of tableaux fixed by evacuation is described by a Motzkin-type formula, revealing deep symmetric and recursive structures.

7. Prime Decomposition and Structural Stability

Both noncrossing involutions and Richardson tableaux admit unique factorizations into prime components—those which cannot be written as a direct sum or tableau concatenation, respectively. Under the bijection given by RS-insertion, prime noncrossing matchings correspond bijectively to prime Richardson tableaux; in this context, prime tableaux are characterized by their shape terminating in two single-box rows (Guo, 19 Nov 2025). This decomposition mirrors algebraic and geometric factorization phenomena observed in Springer theory, strengthening the structural understanding of the subject.

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Noncrossing partial matchings serve as a nexus for enumerative, geometric, and algebraic combinatorics, linking pattern avoidance, lattice path bijections, tableau theory, and computational complexity. Their paper illuminates the interplay of local arc constraints, global structural properties, and deep connections with algebraic objects such as Richardson varieties and Springer fibers (Guo, 19 Nov 2025, Chen et al., 2010, Pilz et al., 2020).