Noncrossing Partitions From Cones and Semicircles

Abstract: For each finite configuration of distinct points in the plane, there is an associated lattice of noncrossing partitions. When these points form the vertices of a convex polygon, the result is the classical noncrossing partition lattice, which is enumerated by the Catalan numbers and satisfies many other useful properties. In this article, we examine three variations of this lattice which arise when the starting configuration is allowed to have points on the sides of a convex polygon rather than just the vertex set.

• The paper establishes that noncrossing partitions in conical and semicircular point sets form graded, bounded lattices with well-defined rank functions.
• It demonstrates symmetric chain decompositions across open cone, closed cone, and semicircular configurations using recursive constructions tied to classical and Boolean lattices.
• The paper derives explicit multivariate generating functions for each configuration, expanding combinatorial tools for analyzing crossing-free structures in planar geometry.

Noncrossing Partition Lattices for Conical and Semicircular Point Configurations

Introduction

The paper "Noncrossing Partitions From Cones and Semicircles" (2604.14462) investigates the combinatorial properties of noncrossing partition lattices, focusing on generalizations beyond classical cases. Traditionally, the lattice of noncrossing partitions ($NC(n)$) is defined for the vertices of a convex $n$-gon and is enumerated by the Catalan numbers, possessing properties such as boundedness, gradedness, self-duality, symmetric chain decomposition, and rank-symmetry. This work extends the study to more complex planar configurations, specifically: points arranged on the sides (not merely the vertices) of a convex polygon, and three particular configuration classes—open cone ($U_{m,n}$), closed cone ($V_{m,n}$), and semicircular ($S_{m,n}$)—each encoding distinct combinatorial and geometric relationships.

Lattice Properties of General Configurations

For a finite set $P$ of points in the plane, a partition is defined as noncrossing if the convex hulls of its blocks are pairwise disjoint. The corresponding poset of noncrossing partitions, $NC(P)$, is a subposet of the full partition lattice $\Pi(P)$, partially ordered by refinement. A principal result demonstrated in this paper is that when $P$ lies on the boundary of a convex polygon (possibly including points on edges as opposed to vertices), $NC(P)$ is always a graded lattice. This is established via an argument on the structure of covering relations in $n$0 and the geometric distribution of points, ensuring that block mergers occur in pairs, guaranteeing a well-defined rank function.

Notably, while the classical cases ($n$1, Boolean lattice for collinear points) are both graded, self-dual, and admit symmetric chain decompositions, more general configurations do not necessarily preserve all these properties. In particular, self-duality is shown to be rare for $n$2 arising from arbitrary boundary configurations.

Three Distinguished Configuration Types

The paper defines and analyzes three fundamental families of planar configurations:

1. Open cone $n$3: $n$4 points distributed on two bounding rays of an affine convex cone, with $n$5 on one ray and $n$6 on the other, excluding the intersection point.
2. Closed cone $n$7: $n$8 points on the boundary, similar to $n$9 with the intersection point included.
3. Semicircular $U_{m,n}$0: $U_{m,n}$1 points on a line segment (the flat side of a semicircle) plus $U_{m,n}$2 points on the semicircular arc.

For each class, the type of lattice $U_{m,n}$3 is determined uniquely by the parameters $U_{m,n}$4, and they are shown to form bounded, graded posets with further refined combinatorial properties.

Symmetric Chain Decomposition and Rank Symmetry

A central contribution of the paper is the demonstration that all three lattice types—$U_{m,n}$5, $U_{m,n}$6, and $U_{m,n}$7—admit symmetric chain decompositions. This is achieved through recursive constructions, leveraging combinations of Boolean and classical noncrossing partition lattices and poset products, ensuring both rank-symmetry and gradedness for these generalized configurations. The symmetric chain decomposition property gives direct control over the rank distribution, confirming that the number of elements at rank $U_{m,n}$8 equals those at rank $U_{m,n}$9 for all relevant $V_{m,n}$0.

However, it is emphasized that, unlike classical cases, these lattices are not generally self-dual, pointing to a nuanced landscape for poset automorphism structures in geometric combinatorics.

Enumerative Results: Generating Functions

The paper derives explicit multivariate generating functions for the cardinality of noncrossing partition lattices in all three configuration families:

• For open cone lattices:

$V_{m,n}$1

• For closed cone lattices:

$V_{m,n}$2

• For semicircular lattices:

$V_{m,n}$3

where $V_{m,n}$4 is the Catalan number generating function, $V_{m,n}$5.

These generating functions encode the recursive structure derived for each configuration class, and for semicircular configurations in particular, the generating function $V_{m,n}$6 interpolates between the classical Catalan enumeration and Boolean lattice counts.

Implications and Future Directions

The combinatorial results have implications for broader mathematical areas, including geometric group theory, Garside structures in Artin groups, and enumerative combinatorics. The explicit structural and enumerative characterization of $V_{m,n}$7 for cone and semicircle configurations expands the toolkit for understanding crossing-free structures in planar geometry and the algebraic properties of posets arising from geometric contexts.

Potential future developments include the investigation of closed enumeration formulas for semicircular lattices, modularity and automorphism groups of these new lattices, and their connections to crossing-free graphs, cell complexes, and other geometric combinatorial objects. Additionally, further exploration into the rare self-duality phenomenon among generalized noncrossing partition lattices may yield deeper classification results.

Conclusion

This paper extends the classical theory of noncrossing partitions to planar point configurations on cones and semicircles, establishing that the resulting lattices are bounded, graded, and rank-symmetric, admitting symmetric chain decompositions but not generally self-dual. The derived multivariate generating functions and recursive decompositions present new enumerative tools for crossing-free partition theory, promising further combinatorial and geometric applications in the study of posets linked to planar configurations.