Noncrossing Partitions From Hull Configurations

Abstract: Each finite configuration of points in the plane determines a corresponding lattice of noncrossing partitions. When these points form the vertex set of a convex polygon, the associated lattice is the classical noncrossing partition lattice (introduced by Kreweras in 1972), which makes many appearances in combinatorics and geometric group theory. If, on the other hand, all points of the configuration lie on a common line segment, the result is a Boolean lattice. In this article, we examine the more general class of hull configurations, which we define to be those which lie either on a line segment or on the boundary of a convex polygon. We prove that the corresponding lattices of noncrossing partitions are unions of maximal Boolean subposets and, under certain circumstances, have symmetric chain decompositions.

• The paper introduces noncrossing partition lattices from hull configurations, revealing a novel framework that unifies classical and Boolean structures.
• It employs combinatorial enumeration and poset analysis to quantify covers, showing that minimal and maximal elements exhibit exponential relationships.
• The study demonstrates that symmetric chain decompositions exist when a blank side is present and establishes bijections with noncrossing trees.

Noncrossing Partition Lattices Arising from Hull Configurations

Introduction and Motivation

The paper "Noncrossing Partitions From Hull Configurations" (2604.14458) investigates the structure and combinatorial properties of noncrossing partition lattices associated with finite planar point configurations, specifically the subclass called hull configurations. Classical noncrossing partition lattices, defined for points in convex position, are central in combinatorics, geometric group theory, and free probability. The Boolean lattice emerges as the structure on collinear point configurations. This work studies lattices associated with configurations interpolating between these extremes—those where all points are either collinear or lie on the boundary of a convex polygon.

Hull Configurations and Partial Orders

Hull configurations are defined as point configurations in the plane whose convex hull contains no interior points. The associated poset $H(n)$ contains all convexity classes with $n$ points, partially ordered by refinement, where elementary collapses correspond to introducing new collinearities. Minimal elements correspond to points on a line segment (Boolean lattices), while maximal elements correspond to points on a convex $n$-gon (classical noncrossing partition lattice $NC(n)$). Intervals in $H(n)$ are isomorphic to Boolean lattices, and $H(n)$ is graded by the number of vertices in the convex hull or the number of collinearities.

The enumeration of elements in $H(n)$ is derived by considering choices of cyclic order and selection of hull vertices. Notably, each minimal element is covered by $2^{n-2}$ maximal elements, and each maximal element covers $n2^{n-3}$ minimal elements, reflecting the combinatorial complexity of these structures.

Noncrossing Partition Lattices $NC(P)$

For each hull configuration $n$0, the lattice $n$1 consists of set partitions such that the convex hulls of blocks are pairwise disjoint. Key properties are inherited or generalized from the classical setting:

• $n$2 is always a lattice [cdhm24].
• For hull configurations, $n$3 is graded [dfjllpr25].
• $n$4 is isomorphic to a subposet of $n$5 in the convex case.

A central focus is the existence and construction of symmetric chain decompositions (SCDs) in $n$6, an important structural property implying rank-symmetry and enabling refined enumerative and algebraic arguments.

Symmetric Chain Decompositions: Main Results

The primary structural theorem states that if $n$7 contains at least one blank side (an edge with only its endpoints among the configuration), then $n$8 admits a symmetric chain decomposition. The argument leverages interval decompositions in the poset, expressing $n$9 as a disjoint union of centered saturated chains and products with smaller $n$0, using inductive constructions and direct products that preserve the SCD property.

The paper conjectures that for hull configurations with no blank sides, $n$1 loses rank-symmetry and cannot admit an SCD, supported by explicit numerical examples showing non-symmetric rank distributions.

Boolean Subposets and Noncrossing Trees

The second main theorem generalizes existing results on the classical lattice, showing that $n$2 for any hull configuration is a union of maximal Boolean subposets, each in bijective correspondence with noncrossing trees on $n$3 that satisfy the convex-geodesic property—a path's convex hull contains only its vertices. This result extends the bijection known for $n$4, where the correspondence is well-studied [bm10, hks16, hypertrees].

The proof is constructive: maximal Boolean subposets correspond to the subforest lattices of noncrossing trees with convex geodesics, and the join structure in these lattices relates directly to the combinatorial properties of the trees and partitions.

The intrinsic geometry of the braid group and its diagonal link has deep connections to these lattices. The embedding properties and union-of-apartments structure have implications for understanding CAT(0) properties of braid groups, especially through the metric and combinatorial simplicial complex constructions tied to $n$5 and its substructures [bm10, jeong23].

Practical and Theoretical Implications

The new class of hull configurations expands the combinatorial landscape where noncrossing partition lattices are tractable and structurally rich. The symmetric chain decomposition results are significant for enumeration, algebraic combinatorics, and applications in group theory, especially for understanding the geometry of associated Coxeter and Artin groups. The Boolean lattice unions facilitate connections to hypertree and building theories, and the combinatorial geometry of planar graphs.

From a theoretical perspective, these results raise new questions regarding SCDs in more general configurations and connections to other poset invariants, such as supersolvability and shellability. Practically, the enumeration and decomposition techniques can impact computational geometry algorithms and problems that involve partitioning datasets in planar contexts.

The implications for the intrinsic geometry of the braid group are noted, with potential for leveraging smaller diagonal links from hull configurations to understand the full $n$6 diagonal link and the associated geometric group theory conjectures.

Conclusion

This paper systematically generalizes classical noncrossing partition lattices, describing the combinatorial and geometric structures arising from hull configurations. Through structural theorems on symmetric chain decompositions and maximal Boolean subposets, it provides a comprehensive framework extending known results, dissecting the lattice-theoretic and geometric-subcomplex properties. The interplay between point configurations, partition lattices, noncrossing trees, and group geometry is elucidated, opening avenues for further research in combinatorics, geometric group theory, and computational geometry.