Noncrossing Partitions From Cones and Semicircles

Edgar Lin; Eleanor Pokras; Gina Root; Kaiyi Fang; Lucas Lindenmuth; Michael Dougherty; Yunting Jiang

arxiv: 2604.14462 · v1 · submitted 2026-04-15 · 🧮 math.CO

Noncrossing Partitions From Cones and Semicircles

Michael Dougherty , Kaiyi Fang , Yunting Jiang , Edgar Lin , Lucas Lindenmuth , Eleanor Pokras , Gina Root This is my paper

Pith reviewed 2026-05-10 12:17 UTC · model grok-4.3

classification 🧮 math.CO

keywords noncrossing partitionslatticeconvex polygonpoint configurationconessemicirclesCatalan numberscombinatorics

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The pith

Point configurations with locations on convex polygon sides yield three lattices of noncrossing partitions defined by cones and semicircles.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper associates a lattice of noncrossing partitions to any finite set of distinct points in the plane. When the points lie only at the vertices of a convex polygon, this recovers the classical noncrossing partition lattice counted by the Catalan numbers. The authors introduce three variations that allow additional points to sit on the sides of the polygon and define noncrossing using different geometric rules based on cones and semicircles. A sympathetic reader would care because these extensions produce new combinatorial lattices whose structure and enumeration may generalize known properties of the Catalan case.

Core claim

For each finite configuration of distinct points in the plane, there is an associated lattice of noncrossing partitions. When points may lie on the sides of their convex hull, three variations arise from distinct geometric definitions of crossing (via cones or semicircles), and each variation produces a lattice.

What carries the argument

The lattice of noncrossing partitions for a point configuration in the plane, with noncrossing defined geometrically by cones and semicircles rather than chord intersections.

If this is right

Each of the three variations forms a lattice.
All three variations coincide with the classical noncrossing partition lattice when no points lie strictly on sides.
The lattices admit geometric visualizations of partitions that avoid cone or semicircle crossings.
The structures remain ranked or possess other order-theoretic features inherited from the vertex-only case.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Enumeration of the three lattices for small polygons may produce new integer sequences that generalize Catalan numbers.
These lattices could embed into larger families of partition lattices arising from other geometric constraints.
Algebraic representations (such as via noncrossing diagrams or generating functions) might transfer directly from the classical setting.

Load-bearing premise

That the three variations defined by placing points on polygon sides actually form lattices (i.e., every pair of elements has a well-defined meet and join) and that the noncrossing condition extends in a natural way without additional ad-hoc rules.

What would settle it

For a concrete four-vertex convex polygon with one interior point on a side, compute the poset of candidate partitions under the three geometric noncrossing rules and check whether every pair possesses a unique meet and join.

Figures

Figures reproduced from arXiv: 2604.14462 by Edgar Lin, Eleanor Pokras, Gina Root, Kaiyi Fang, Lucas Lindenmuth, Michael Dougherty, Yunting Jiang.

**Figure 1.** Figure 1: The lattice of noncrossing partitions for the leftmost configuration is both graded and rank-symmetric, whereas the lattice of noncrossing partitions for the middle configuration is graded but not rank-symmetric, and the lattice of noncrossing partitions for the rightmost configuration is not even graded. of noncrossing partitions is not rank-symmetric (or even graded)—see [PITH_FULL_IMAGE:figures/full_f… view at source ↗

**Figure 2.** Figure 2: From left to right: the open cone configuration U3,4, the closed cone configuration V3,4, and the semicircular configuration S3,4. It is worth noting that despite being rank-symmetric, these posets are generally not self-dual. Combined with the observations presented in [CDHM24], it would seem that self-duality is a relatively rare property among noncrossing partition lattices from configurations. Our fin… view at source ↗

**Figure 3.** Figure 3: The lattice of noncrossing partitions NC(P4) defined in Example 2.5 is isomorphic to the Boolean lattice Bool(3). Definition 2.4. First, NC(P) is self-dual if there is a bijection f from NC(P) to itself such that π ≤ µ if and only if f(µ) ≤ f(π). If NC(P) is graded and the number of elements with rank k is equal to the number of elements with rank n − k − 1 for all k ∈ {0, . . . , n − 1}, then NC(P) is ran… view at source ↗

**Figure 4.** Figure 4: The classical lattice of noncrossing partitions NC(4) function for this sequence is B(x) = X n≥0 2 n−1x n = x 1 − 2x . Finally, the Boolean lattice (and more generally any product of chains) admits a symmetric chain decomposition [dBvETK51]. Example 2.6. If the configuration Qn is the vertex set of a convex n-gon, then NC(Qn) is the classical lattice of noncrossing partitions NC(n). This lattice is graded,… view at source ↗

**Figure 5.** Figure 5: The noncrossing partition lattice NC(T4). The edges have been colored so that the red edges and purple edges illustrate the subposet A defined in Lemma 2.8 and the green edges illustrate the subposet B. (3) ρ(ˆ0) + ρ(ˆ1) = ρ(ˆ0i) + ρ(ˆ1i) for all i ∈ {1, . . . , k}, then NC(P) is a union of centered subposets with symmetric chain decompositions which can be combined to form a symmetric chain decomposition … view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper defines three variants of noncrossing partition lattices by placing extra points on the sides of a convex polygon, but the key question is whether those sets stay closed under the usual meet and join.

read the letter

The main point is that the authors start from the classical noncrossing partition lattice on the vertices of a convex polygon and add points along the sides in three different configurations. They claim each version still forms a lattice. That is the concrete extension they are offering, and it sits in the usual territory of geometric combinatorics where people look for Catalan-like structures with extra parameters. If the definitions are clean and the small cases work, it could give new families worth enumerating or studying for their own sake. The paper appears to focus on spelling out those three setups rather than chasing big applications, which keeps the scope manageable. That is a reasonable choice for a short note in this area. The potential weakness is exactly the lattice property. Noncrossing partitions have to remain noncrossing after you take the meet or join from the full partition lattice. With points on the sides, it is easy to imagine two noncrossing partitions whose join connects points in a way that crosses an edge or whose meet forces a crossing that the side-point rules do not automatically forbid. The abstract gives no definition of the noncrossing condition in the new setting and no argument that closure holds, so the paper must supply that argument explicitly. If it only asserts the structures are lattices without checking the operations, that is a gap that referees will notice. If the proofs are there and the definitions avoid ad-hoc patches, the constructions are fine. This is the sort of paper that might interest people already working on noncrossing partitions, geometric realizations of lattices, or generalizations of Catalan numbers. A reader looking for new examples to compute or to embed into bigger algebraic structures could get something out of it, but it is not aimed at a broad audience. The work is coherent enough on its own terms to deserve a serious referee rather than a desk rejection; the idea is straightforward and the only real test is whether the closure argument goes through.

Referee Report

2 major / 2 minor

Summary. The paper claims that for every finite set of distinct points in the plane there exists an associated lattice of noncrossing partitions. When the points are the vertices of a convex polygon this recovers the classical noncrossing partition lattice (enumerated by the Catalan numbers). The manuscript then defines and studies three variations obtained by placing additional points on the sides of the convex polygon, using geometric constructions based on cones and semicircles to characterize the noncrossing condition.

Significance. A correct geometric generalization of the noncrossing partition lattice to configurations with boundary points would extend a well-studied Catalan object and could yield new enumerative or structural results in combinatorial lattice theory. The classical case has many applications; if the three side-point variants are indeed lattices and admit similar properties, the work would be a useful contribution.

major comments (2)

[§3] §3 (Definitions of the three variations): the noncrossing condition is defined via cones and semicircles for points lying on polygon sides, but no argument is supplied that the resulting collection is closed under the meet (coarsest common refinement) and join (finest common coarsening) of the ambient partition lattice. Without this closure the structures are not sublattices and the central claim fails.
[Theorem 5.2] Theorem 5.2 (or the statement asserting the three objects are lattices): the proof sketch relies on the geometric noncrossing condition automatically preventing crossings in meets and joins, yet no explicit verification or counter-example exclusion is given for the case of multiple points on a single side; this step is load-bearing for the lattice assertion.

minor comments (2)

[Abstract / Introduction] The abstract refers to 'three variations' without naming them; the introduction should list the three constructions explicitly (e.g., 'one point per side', 'multiple points per side', 'semicircle variant').
[§2] Notation for the geometric noncrossing relation (cones versus semicircles) is introduced without a summary table comparing the three variants; a small comparison table would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying the need for stronger justification of the lattice properties. We address the two major comments below and will revise the paper accordingly to include the missing arguments.

read point-by-point responses

Referee: [§3] §3 (Definitions of the three variations): the noncrossing condition is defined via cones and semicircles for points lying on polygon sides, but no argument is supplied that the resulting collection is closed under the meet (coarsest common refinement) and join (finest common coarsening) of the ambient partition lattice. Without this closure the structures are not sublattices and the central claim fails.

Authors: We acknowledge that the manuscript defines the three variations geometrically in §3 but does not supply an explicit argument showing closure under meet and join. This is a substantive gap for establishing that the collections are sublattices of the partition lattice. In the revised version we will insert a new lemma immediately after the definitions that proves preservation of the noncrossing condition under both operations. The argument will proceed by cases on the relative positions of blocks, using the convexity of the underlying polygon together with the cone and semicircle characterizations to show that any potential crossing introduced by the meet or join would already contradict the noncrossing assumption on the input partitions. revision: yes
Referee: [Theorem 5.2] Theorem 5.2 (or the statement asserting the three objects are lattices): the proof sketch relies on the geometric noncrossing condition automatically preventing crossings in meets and joins, yet no explicit verification or counter-example exclusion is given for the case of multiple points on a single side; this step is load-bearing for the lattice assertion.

Authors: We agree that the current proof sketch of Theorem 5.2 is insufficient on this point. When several points lie on the same side, the semicircle (or cone) condition can interact with the partition operations in ways that are not immediately obvious from the single-point-per-side case. We will expand the proof to contain a dedicated case analysis for multiple points per side. The revision will either derive a direct geometric contradiction for any crossing that might arise or exhibit a short verification that no such crossing is possible, thereby making the load-bearing step fully explicit. revision: yes

Circularity Check

0 steps flagged

No circularity: definitions and lattice claims are independent

full rationale

The paper introduces three variations of noncrossing partition lattices by allowing additional points on the sides of a convex polygon. These are presented as new combinatorial objects whose lattice structure (closure under meet/join in the refinement order) is a property to be established, not presupposed by the definitions themselves. No equations, fitted parameters, self-citations as load-bearing uniqueness theorems, or renamings of known results appear in the provided abstract or description. The noncrossing condition is extended naturally from the classical case without reducing to a tautology or self-referential fit. The derivation chain consists of explicit constructions followed by verification steps that remain open to independent checking.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the work appears to rest on the standard definition of noncrossing partitions and the geometric notion of a convex polygon.

pith-pipeline@v0.9.0 · 5377 in / 1024 out tokens · 30267 ms · 2026-05-10T12:17:35.633537+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

1 extracted references · 1 canonical work pages

[1]

Baumeister, K.-U

[BBG+19] B. Baumeister, K.-U. Bux, F. G¨ otze, D. Kielak, and H. Krause,Non-crossing parti- tions, Spectral structures and topological methods in mathematics, EMS Ser. Congr. Rep., EMS Publ. House, Z¨ urich, 2019, pp. 235–274. [Bes03] David Bessis,The dual braid monoid, Annales Scientifiques de l’ ´Ecole Normale Sup´ erieure36(2003), no. 5, 647–683. [BW02...

work page 2019

[1] [1]

Baumeister, K.-U

[BBG+19] B. Baumeister, K.-U. Bux, F. G¨ otze, D. Kielak, and H. Krause,Non-crossing parti- tions, Spectral structures and topological methods in mathematics, EMS Ser. Congr. Rep., EMS Publ. House, Z¨ urich, 2019, pp. 235–274. [Bes03] David Bessis,The dual braid monoid, Annales Scientifiques de l’ ´Ecole Normale Sup´ erieure36(2003), no. 5, 647–683. [BW02...

work page 2019