Noncrossing Partition Lattices from Planar Configurations

Abstract

The lattice of noncrossing partitions is well-known for its wide variety of combinatorial appearances and properties. For example, the lattice is rank-symmetric and enumerated by the Catalan numbers. In this article, we introduce a large family of new noncrossing partition lattices with both of these properties, each parametrized by a configuration of n points in the plane.

1 Introduction

For each subset P of the complex plane, a noncrossing partition of P is a way of dividing P into subsets with pairwise disjoint convex hulls. The collection of all noncrossing partitions of P, denoted $\textsc {NC}(P)$, is a partially ordered set under refinement. When P is the vertex set for a convex n-gon, $\textsc {NC}(P)$ is the classical noncrossing partition lattice $\textsc {NC}_n$ introduced by Kreweras [3]. Among other things, Kreweras showed that the size of $\textsc {NC}_n$ is counted by the combinatorially ubiquitous Catalan numbers $C_n = \frac{1}{n+1}\left( {\begin{array}{c}2n\\ n\end{array}}\right)$ and, more specifically, the number of lattice elements with rank k is the Narayana number $N_{n,k} = \frac{1}{n} \left( {\begin{array}{c}n\\ k\end{array}}\right) \left( {\begin{array}{c}n\\ k-1\end{array}}\right)$. Since $N_{n,k} = N_{n,n-k}$, this further says that $\textsc {NC}_n$ is a rank-symmetric lattice. In the fifty years since its definition, the noncrossing partition lattice has made countless appearances in algebraic and geometric combinatorics - see the survey articles [4] and [1] for more information.

Returning to the more general case prompts a natural question: for which subsets $P \subset \mathbb {C}$ does the poset $\textsc {NC}(P)$ have similar properties to $\textsc {NC}_n$? While some existing work studies the size of $\textsc {NC}(P)$ in asymptotic and extremal cases (e.g. [7] [6]), similarities to $\textsc {NC}_n$ seem uncommon in the literature. In our first main theorem, we introduce a convexity condition on P which guarantees that $\textsc {NC}(P)$ has the same size as $\textsc {NC}_n$.

Let $P \subset \mathbb {C}$ be a set of n points in general position. We say that a subset $A\subseteq P$ is convex if no point in A lies in the convex hull of the others, and we say that P satisfies Property $\Delta _k$ if, for all convex $A\subseteq P$, the convex hull of A contains at most $|A| + k - 3$ elements of P in its interior.

Theorem A

(Theorem 5.4) Let $P \subset \mathbb {C}$ be a set of n points with Property $\Delta _1$. Then $\textsc {NC}(P)$ is a rank-symmetric graded lattice, and the number of elements with rank k is the Narayana number $N_{n,k}$. In particular, $|\textsc {NC}(P)| = C_n = |\textsc {NC}_n|$.

It is worth noting that if P contains a point which lies in the convex hull of the others, then $\textsc {NC}(P)$ is not isomorphic to $\textsc {NC}_n$. Thus, Theorem A introduces a large new class of lattices with the same number of elements in each rank as the noncrossing partition lattice.

If the conditions on P are weakened to only require Property $\Delta _2$, then the number of noncrossing partitions may increase compared to those in Theorem A. Nevertheless, some symmetry is preserved.

Theorem B

(Theorem 6.4) Let $P \subset \mathbb {C}$ be a set of n points with Property $\Delta _2$. Then $\textsc {NC}(P)$ is a rank-symmetric graded lattice.

See Figs. 1 and 2 for examples of configurations which satisfy Properties $\Delta _1$ and $\Delta _2$ respectively. One can quickly find examples of P without Property $\Delta _1$ such that $\textsc {NC}(P)$ fails to be rank-enumerated by the Narayana numbers and examples of P without Property $\Delta _2$ such that $\textsc {NC}(P)$ fails to be rank-symmetric. In fact, $\textsc {NC}(P)$ can even fail to be graded if P does not have Property $\Delta _2$ (see Proposition 4.4 and Fig. 13). The extent to which these properties are necessary conditions for Theorems A and B is not immediately clear; this is an interesting direction for future research.

Some of the techniques used in proving Theorems A and B can be interpreted in a stronger topological context. Recall that the (unordered) configuration space of n points in $\mathbb {C}$ is the topological space of all n-tuples in $\mathbb {C}^n$ with distinct entries, considered up to permutations of the coordinates. Also, if P barely fails to be in general position (i.e. there is a single triple of collinear points in P) but otherwise satisfies Property $\Delta _k$, we say that P satisfies the weak Property $\Delta _k$.

Fig. 1

Some examples of configurations which satisfy Property $\Delta _1$

Fig. 2

Some examples of configurations which satisfy Property $\Delta _2$

Theorem C

(Corollary 3.11) Let $k\in \{1,2\}$. The set of all configurations which satisfy the weak Property $\Delta _k$ forms a connected subspace of the configuration space of n points in $\mathbb {C}$.

We are unaware of any prior appearances of the space described in Theorem C. It would be interesting to know the homology of this space for each k and, in particular, whether it is a classifying space for the n-strand braid group.

The article is structured as follows. In Sect. 2, we introduce some background on posets and partitions, along with basic properties of $\textsc {NC}(P)$. Section 3 concerns the transformation of configurations with Property $\Delta _k$ and includes the proof of Theorem C. We provide some technical details on “skewers” in Sect. 4, then give the proofs of Theorem A in Sect. 5 and Theorem B in Sect. 6.

2 Noncrossing Partitions

To start, we establish some basic definitions and properties for partitions, posets, and configurations - see [9] for a standard reference. Recall that a partition expresses a set S as the union of a collection of pairwise disjoint subsets of S (called blocks). The set of all partitions for a fixed set S forms a partially ordered set under refinement: one partition lies “below” another in the partial order if each block in the latter partition can be obtained as a union of blocks in the former. This partially ordered set is a lattice in the sense that each pair of elements has a unique meet (greatest lower bound) and a unique join (least upper bound). Let $\Pi (S)$ denote the lattice of partitions for S; in the standard case where $S = \{1,\ldots ,n\}$, the associated partition lattice is denoted $\Pi _n$.

The partition lattice $\Pi (S)$ is bounded in the sense that it contains a unique minimum element ${\hat{0}}$ (in which each block is a singleton) and a unique maximum element ${\hat{1}}$ (in which all of S belongs to the same block). The partition lattice is also graded: if $|S| = n$ and we let $bl(\pi )$ denote the number of blocks in a partition $\pi \in \Pi (S)$, then the map $\rho :\Pi (S) \rightarrow \mathbb {N}$ given by $\rho (\pi ) = n - bl(\pi )$ is a rank function for this lattice. Note that the minimum ${\hat{0}}$ and maximum ${\hat{1}}$ have ranks 0 and $n-1$ respectively. The atoms and coatoms of this lattice are defined to be the elements of rank 1 and $n-2$ respectively.

Our main object of study in this article is a subposet of the partition lattice for a finite set of points in the complex plane.

Definition 2.1

Fix $P\subset \mathbb {C}$ finite. For any $A \subseteq P$, the convex hull of A, denoted $\textsc {Conv}(A)$, is the smallest convex subset of $\mathbb {C}$ which contains A. Note that $\textsc {Conv}(A)$ is a convex polygon with up to |A| vertices. A partition of P is noncrossing if the convex hulls of its blocks are pairwise disjoint. The set of all noncrossing partitions for P forms a subposet of the partition lattice $\Pi (P)$, and we refer to this subposet as $\textsc {NC}(P)$.

Example 2.2

Let $P = \{z_1,z_2,z_3,z_4\}$ be a set of points in $\mathbb {C}$ such that $z_1$, $z_2$, and $z_3$ form the vertices of a triangle which contains $z_4$ in its interior. Then every partition of P is noncrossing except for $\{\{z_1,z_2,z_3\},\{z_4\}\}$, so the noncrossing partition lattice $\textsc {NC}(P)$ has 14 elements, arranged according to the diagram in Fig. 3. Note that $\textsc {NC}(P)$ has the same size as the classical noncrossing partition lattice $\textsc {NC}_4$, but the two are not isomorphic since $\textsc {NC}(P)$ has 15 maximal chains, while $\textsc {NC}_4$ has 16.

Fig. 3

The lattice of noncrossing partitions for a particular arrangement of four points in $\mathbb {C}$

As a poset, the noncrossing partitions of P inherit some useful properties from the larger partition lattice $\Pi (P)$.

Proposition 2.3

If $P\subset \mathbb {C}$ is finite, then $\textsc {NC}(P)$ is a bounded lattice.

Proof

Since the minimum and maximum elements of $\Pi (P)$ are noncrossing, we know that $\textsc {NC}(P)$ is bounded. To show that $\textsc {NC}(P)$ is a lattice, we need only prove that $\textsc {NC}(P)$ is a meet-semilattice (i.e. that each pair of elements in $\textsc {NC}(P)$ has a unique meet) by a standard property of finite bounded posets [9, Prop. 3.3.1]. First, suppose that $\pi \le \pi '$ in $\Pi (P)$. If $\pi '$ is noncrossing but $\pi$ has a pair of crossing blocks $A,B \in P$, then $\pi '$ must have a block which contains the union $A\cup B$. It follows that if $\pi _1,\pi _2\in \textsc {NC}(P)$, then the meet $\pi _1 \wedge \pi _2$ must be noncrossing; if it had a pair of crossing blocks A and B, then $\pi _1$ and $\pi _2$ would each have blocks containing $A\cup B$ by the argument above and thus $\pi _1 \wedge \pi _2$ would have a block containing $A\cup B$, a contradiction. Therefore, $\textsc {NC}(P)$ is a meet-semilattice and thus a lattice. $\square$

Although $\Pi (P)$ is graded for every configuration P, it is possible to construct P so that the poset $\textsc {NC}(P)$ is not graded. In this article, however, we are interested in configurations satisfying particular conditions (Properties $\Delta _1$ and $\Delta _2$, defined in Sect. 3), and we will see in Sect. 4 that $\textsc {NC}(P)$ is graded in these cases.

We close the section with a few important examples and some observations.

Example 2.4

If $P \subset \mathbb {C}$ with $|P| = n$ such that each point in P lies on the boundary of $\textsc {Conv}(P)$ (i.e. P is in convex position), then $\textsc {NC}(P)$ is isomorphic to the classical noncrossing partition lattice $\textsc {NC}_n$, initially defined by Kreweras [3] - see Fig. 4. In addition to the properties outlined in Proposition 2.3, Kreweras showed that the size of $\textsc {NC}_n$ is equal to the Catalan number $C_n = \frac{1}{n+1}\left( {\begin{array}{c}2n\\ n\end{array}}\right)$ and, in particular, the number of partitions in $\textsc {NC}_n$ with k blocks is the Narayana number $N_{n,k} = \frac{1}{n} \left( {\begin{array}{c}n\\ k\end{array}}\right) \left( {\begin{array}{c}n\\ k-1\end{array}}\right)$. For more information on the combinatorial significance of these connections, see [10].

Fig. 4

The classical noncrossing partition lattice $\textsc {NC}_4$

Noting that $N_{n,k} = N_{n,n-k}$ for all $1\le k \le n$, one can see that $\textsc {NC}_n$ is a rank-symmetric lattice. In fact, the classical noncrossing partition lattice is self-dual in the sense that it admits a bijection $\alpha :\textsc {NC}_n \rightarrow \textsc {NC}_n$ with the property that $\pi _1 \le \pi _2$ if and only if $\alpha (\pi _2) \le \alpha (\pi _1)$ [8]. However, this stronger condition is rarely held by $\textsc {NC}(P)$ more generally.

Remark 2.5

Nica and Speicher showed in 1997 that intervals in the noncrossing partition lattice $\textsc {NC}_n$ are isomorphic to products of smaller noncrossing partition lattices [5]. With this in mind, we note that if P is a set of n points of $\mathbb {C}$ in general position and if some point of P lies in the convex hull of the others, then $\textsc {NC}(P)$ has an interval which is isomorphic to the lattice described in Example 2.2, which cannot be expressed as a product of noncrossing partition lattices. Therefore, $\textsc {NC}(P)$ is only isomorphic to $\textsc {NC}_n$ if the elements of P form the vertices of a convex n-gon.

Example 2.6

If $P\subset \mathbb {C}$ with $|P| = n$ such that all points in P are collinear, then $\textsc {NC}(P)$ is isomorphic to the Boolean lattice ${\textsc {Bool}}_{n-1}$, which is defined as the set of all subsets of a set with $n-1$ elements, partially ordered under inclusion. To see this, observe that each partition in $\textsc {NC}(P)$ is determined precisely by choosing a subset of the $n-1$ gaps between the n points; see Fig. 5.

Fig. 5

The Boolean lattice ${\textsc {Bool}}_{n-1}$ arises as the set of noncrossing partitions for a configuration of n collinear points

When $|P|=4$, there are only four possibilities for $\textsc {NC}(P)$ (up to isomorphism), and three of them are depicted in the preceding figures. All three (indeed, all four) possess several useful lattice properties, including rank-symmetry, self-duality, and a simple counting formula. However, these properties do not always hold for larger sizes of P.

Example 2.7

If P consists of five points in general position with three points on the boundary of the convex hull and two points in the interior, then $|\textsc {NC}(P)| = 43$ (whereas $|\textsc {NC}_n| = 42$), although $\textsc {NC}(P)$ remains rank-symmetric. Furthermore, if P consists of six points in general position, arranged so that the three extremal points form an equilateral triangle and the three interior points form a shrunken equilateral triangle with the same center, then $\textsc {NC}(P)$ is not rank-symmetric: it has 15 atoms (rank 1) and coatoms (rank 4), but 55 elements at rank 2 and 57 elements at rank 3.

3 Configurations

Before moving on to the main theorems, we introduce some tools for studying the geometry of planar configurations, by which we mean finite unordered sets of points in the Euclidean plane. We begin with some helpful terminology, partially inspired by [2]. Throughout this section, let P denote a configuration of n points in $\mathbb {C}$ in general position (i.e. no three points in P are collinear), unless otherwise specified.

Definition 3.1

Let $A \subseteq P$ and recall that $\textsc {Conv}(A)$ denotes the convex hull of the points in A. Define the closure ${\overline{A}}$ by $\textsc {Conv}(A) \cap P$ and the interior $\text {int}(A)$ to be $\text {int}(\textsc {Conv}(A)) \cap P$. We say that A is convex if $\text {int}(A)\cap A$ is empty. Also, a point $p\in P$ is internal if $p\in \text {int}(P)$ and extremal otherwise.

Definition 3.2

A configuration P in general position satisfies Property $\Delta _k$ if, for every convex subset A in P, the interior $\text {int}(A)$ contains at most $(|A|-3) + k$ points. Equivalently, P has Property $\Delta _k$ if P is in general position and each subset $B\subseteq P$ (not necessarily convex) has at most $\lfloor \frac{|B|-3+k}{2} \rfloor$ internal points. Finally, we say that P instead has the weak Property $\Delta _k$ if it satisfies the same convexity criteria, but has at most one instance of three collinear points.

It is worth noting that Property $\Delta _1$ is equivalent to a simpler condition which is easier to check: for any $A\subseteq P$ with $|A|=3$, we have $|\text {int}(A)| \le 1$. To see this, consider that each convex subset of m points in P forms the vertices of a convex m-gon, and any triangulation of this polygon consists of $m-2$ triangles; Property $\Delta _1$ is equivalent to the requirement that each of those $m-2$ triangles has at most one point of P in its interior. Unfortunately, this does not generalize to Property $\Delta _k$ when $k > 1$.

If P satisfies Property $\Delta _k$, then any small perturbation of P will also satisfy Property $\Delta _k$ since P is in general position. However, deformations which move a point in P across the line between two other points in P might not preserve this property. The main goal of this section is to provide some tools for moving points in P while preserving Property $\Delta _k$. To start, we establish some terminology for the lines connecting points in P.

Definition 3.3

Each pair of distinct points in P determines a line in $\mathbb {C}$; let $\mathcal {A}$ denote the arrangement of the $\left( {\begin{array}{c}n\\ 2\end{array}}\right)$ lines obtained in this way. If $\ell$ is the line obtained from the points z and w in P, then we say that z and w are the endpoints of $\ell$ and write $V(\ell ) = \{z,w\}$. We also refer to the line segment between z and w as the core $c(\ell ) = \textsc {Conv}(V(\ell ))$ of $\ell$. More generally, we write $V(\ell _1,\ldots ,\ell _k)$ to mean the 2k-element set of endpoints belonging to the lines $\ell _1,\ldots ,\ell _k$ and we write $c(\ell _1,\ldots ,\ell _k)$ to mean the convex hull $\textsc {Conv}(V(\ell _1,\ldots ,\ell _k))$.

Definition 3.4

For each z in $\text {int}(P)$, let $\mathcal {A}^z$ denote the subarrangement of $\mathcal {A}$ obtained by deleting the lines which pass through z. Also, define $\mathcal {A}^{\text {ex}}$ to be the subarrangement of $\mathcal {A}$ which consists of all lines with two extremal endpoints, i.e. the intersection of all $\mathcal {A}^z$ for $z\in \text {int}(P)$. We associate two regions to each $z\in \text {int}(P)$: the connected component of $\mathbb {C}-\mathcal {A}^z$ containing z (which we denote $R_z$) and the connected component of $\mathbb {C} - \mathcal {A}^{\text {ex}}$ containing z (denoted $R_z^\text {ex}$). Note that each region is a convex polygon since it is a bounded subset of the plane determined by removing some number of half-planes. Finally, we say that a line in $\mathcal {A}$ is separating if it has points from $\text {int}(P)$ on either side of it, and a boundary line is one which contains a boundary edge for the convex hull $\textsc {Conv}(P)$.

For the sake of clarity, we will typically illustrate the line arrangement $\mathcal {A}$ by its intersection with the convex hull of P - see Fig. 6 for an example.

Fig. 6

From left to right: an arrangement $\mathcal {A}$, the subarrangement $\mathcal {A}^z$ with the region $R_z$ highlighted, and the subarrangement $\mathcal {A}^{\text {ex}}$ of lines between pairs of extremal points. In each image, only the core of each line has been drawn

Definition 3.5

A move is a bijection $m:P \rightarrow m(P)$ such that m fixes all of P except some element z, which is instead sent to a point m(z) in the interior of a region adjacent to $R_z$. If $\ell$ is a line in the arrangement $\mathcal {A}$ which separates the regions $R_z$ and $R_{m(z)}$, then we say that m moves z across $\ell$. If both P and m(P) satisfy Property $\Delta _k$, we say that m is a $\Delta _k$-move. Finally, note that m induces an isomorphism $m_*:\Pi (P) \rightarrow \Pi (m(P))$ by replacing z with m(z) in each partition.

It is worth noting that for any $z\in P$, we can replace z with any other point in the region $R_z$ without changing the isomorphism type of $\Pi (P)$, so moves on P can be described solely by the regions involved.

Definition 3.6

Let $z\in P$. If $\ell$ is a line in the arrangement $\mathcal {A}$ which contains a side of the region $R_z$, then we say that $\ell$ is adjacent to z. This line determines two open half-planes: $H^{+}_{z,\ell }$, which contains z, and $H^-_{z,\ell }$, which does not.

Fig. 7

If $m:P \rightarrow m(P)$ is a move which takes z across a non-separating edge between extremal points $w_1$ and $w_2$, and if the triple $\{a_1,a_2,m(z)\}$ (indicated with dashed lines) has more than one point in its interior, then the quadrilateral $\{a_1,a_2,w_1,w_2\}$ (indicated with solid lines) has at least three points in its interior

The following lemmas establish two useful cases of $\Delta _k$-preserving moves.

Lemma 3.7

If P has Property $\Delta _k$ and $m:P \rightarrow m(P)$ moves $z\in \text {int}(P)$ across a non-separating line in $\mathcal {A}^{\text {ex}}$, then m(P) has Property $\Delta _k$ as well.

Proof

Let $m:P \rightarrow m(P)$ be a move which takes z across a line $\ell$ in $\mathcal {A}^{\text {ex}}$, and let $w_1$ and $w_2$ be the endpoints of $\ell$. Suppose for the sake of contradiction that m(P) does not satisfy Property $\Delta _k$; then there is a subset $A \subseteq m(P)$ with $|\text {int}(A)| > \lfloor \frac{|A|-3+k}{2}\rfloor$. Since P satisfies Property $\Delta _k$, we know that ${\overline{A}}$ must contain m(z) but not z.

First, note that the supposed bound on $|\text {int}(A)|$ precludes m(z) from being an internal point of A. If it were internal, then A would necessarily contain $w_1$ and $w_2$, and A would therefore be a subset of $H^{+}_{m(z),\ell }$ since $z \not \in {\overline{A}}$. However, m(z) is the unique internal point of m(P) in $H^{+}_{m(z),\ell }$ since we assumed $\ell$ was non-separating in the initial configuration P, so the inequality would not hold.

Thus, m(z) is an extremal point of A. Define $B = (A-\{m(z)\})\cup \{z,w_1,w_2\}$; then $|B| \le |A| + 2$ and $|\text {int}(B)| \ge |\text {int}(A)| + 1$ (since z is an internal point for B but not A), and we can combine these to find the following chain of inequalities:

$$\begin{aligned} |\text {int}(B)|\ge & {} |\text {int}(A)| + 1 > \left\lfloor \frac{|A|-3+k}{2}\right\rfloor + 1\\ {}\ge & {} \left\lfloor \frac{|A|-3+k+2}{2}\right\rfloor \ge \left\lfloor \frac{|B|-3+k}{2}\right\rfloor . \end{aligned}$$

Since B is a subset of P, this contradicts our assumption that P has Property $\Delta _k$ - see Fig. 7 for an example when $k=1$. Therefore, m(P) must satisfy Property $\Delta _k$ and we are done. $\square$

Fig. 8

If $w_2$ is an internal point and m is a move which takes z across the line containing $w_1$ and $w_2$, then there are extremal points $a_1$ and $a_2$ such that the convex hull of $a_1$, $a_2$, and $w_1$ (depicted with dashed lines) has $w_2$, z, and m(z) in its interior

Lemma 3.8

If P has Property $\Delta _k$ and $m:P \rightarrow m(P)$ moves $z\in \text {int}(P)$ across a line in $\mathcal {A}^z$ with at least one internal endpoint, then m(P) has Property $\Delta _k$ as well.

Proof

Let $\ell$ be a line in $\mathcal {A}^z$ adjacent to z with endpoints $w_1$ and $w_2$, suppose that $w_2$ is an internal point of P, and let $m:P\rightarrow m(P)$ be the move which takes z across $\ell$. As above, we suppose for the sake of contradiction that m(P) does not satisfy Property $\Delta _k$ and can thus find a subset $A\subseteq m(P)$ with $|\text {int}(A)| > \lfloor \frac{|A|-3+k}{2}\rfloor$ such that ${\overline{A}}$ contains m(z) but not z.

Consider the three lines in $\mathcal {A}$ for which one endpoint is $w_1$ and the other belongs to the set $\{z,w_2,m(z)\}$. Since z, $w_2$, and m(z) are internal points of P and both z and m(z) are adjacent to $\ell$, all three of these lines must pass through the same side of the polygon $\textsc {Conv}(P)$. Let $a_1,a_2\in P$ be the extremal points which determine this side, where $a_1$ is on the same side of $\ell$ as z - see Fig. 8 for an illustration.

Now, define $B = (A-\{m(z)\}) \cup \{z,w_1,w_2,a_1,a_2\}$. If $m(z)\in \text {int}(A)$, then A must contain $w_1$ and $w_2$, and the fact that $z\notin A$ tells us that A does not contain any points in the half-plane $H^{+}_{z,\ell }$. Thus, in this case we have that $|B| \le |A| + 2$ and $|\text {int}(B)| \ge |\text {int}(A)| + 1$ (since $w_2$ is internal for B but not A), which provides the same chain of inequalities as described in the proof of Lemma 3.7. Therefore, P does not satisfy Property $\Delta _k$, which is a contradiction.

If m(z) is instead an extremal point of A, then we see that $|B| \le |A| + 4$ and $|\text {int}(B)| \ge |\text {int}(A)| + 2$, so we have a similar sequence of inequalities:

$$\begin{aligned} |\text {int}(B)|\ge & {} |\text {int}(A)| + 2 > \left\lfloor \frac{|A|-3+k}{2}\right\rfloor + 2\\ {}\ge & {} \left\lfloor \frac{|A|-3+k+4}{2}\right\rfloor \ge \left\lfloor \frac{|B|-3+k}{2}\right\rfloor . \end{aligned}$$

This also contradicts our assumption that P satisfies Property $\Delta _k$, so we conclude that m(P) must satisfy Property $\Delta _k$. $\square$

Fig. 9

This configuration of 15 points has five internal points for which the corresponding regions have a convex hull (outlined with dashed blue lines in the upper left image) where no side of the convex hull contains a side of a region. By extending each side of the convex hull into a line, each of the five points determines a cone (shaded red in the upper right image) which contains at least one extremal point. Selecting one extremal point from each cone (highlighted red in the bottom image) yields a set of extremal points which contains the starting internal points and is at most as numerous, thus violating Property $\Delta _2$

Note that the following lemma supposes only that P has Property $\Delta _2$, so in particular it holds when P satisfies Property $\Delta _1$ as well.

Lemma 3.9

If P has Property $\Delta _2$, then there is a point $z\in \text {int}(P)$ such that at least one side of the region $R_z^\text {ex}$ belongs to a non-separating line in $\mathcal {A}^{\text {ex}}$.

Proof

Let P be a configuration satisfying Property $\Delta _2$ and let D be the convex hull of the regions $R_z^\text {ex}$, where z is an interior point of P. Let $\ell _1,\ldots ,\ell _k$ denote the lines (not necessarily in $\mathcal {A}$) which contain the k sides of D, arranged so that they appear in counter-clockwise order. If at least one $\ell _i$ contains a side of some region $R_z^\text {ex}$, then $\ell _i$ belongs to the arrangement $\mathcal {A}^{\text {ex}}$, and it follows that $\ell _i$ must be non-separating since all internal points lie on one side of it.

Suppose for the sake of contradiction that this is not the case. Then we can fix points $z_1,\ldots ,z_k \in \text {int}(P)$ such that for each i, the region $R_{z_i}$ intersects the boundary of D at the point where the lines $\ell _i$ and $\ell _{i+1}$ (evaluated mod k) intersect. Since neither $\ell _i$ nor $\ell _{i+1}$ are in $\mathcal {A}^{\text {ex}}$, it follows that the half-planes $H^-_{z_i,\ell _i}$ and $H^-_{z_i,\ell _{i+1}}$ intersect in an unbounded region which contains an extremal point of P - see Fig. 9 for an illustration.

If E is the set of k extremal points obtained in the manner above, then one can show that the convex hull $\textsc {Conv}(E)$ contains D. However, this means that $z_1,\ldots ,z_k$ lie in the interior of E, which contradicts our assumption that P satisfies Property $\Delta _2$. Therefore, at least one side of D must belong to a non-separating line in $\mathcal {A}^{\text {ex}}$ and we are done. $\square$

Putting all of these tools together, we obtain a useful connectivity property.

Theorem 3.10

Suppose P satisfies Property $\Delta _k$, where $k\in \{1,2\}$. Then there is a sequence of $\Delta _k$-moves which transforms P into a convex configuration.

Proof

We proceed by induction on the number of internal points for P. If P has no internal points, then P is already convex and we are done. Now, suppose the theorem holds for configurations with fewer internal points than P and define $\text {nsnb}(P)$ to be the set of non-separating non-boundary lines in $\mathcal {A}^{\text {ex}}$; we will prove that the theorem holds for P as well by a second induction on $|\text {nsnb}(P)|$. If $|\text {nsnb}(P)| = 0$, then each internal point of P is adjacent to a boundary line, and by Lemma 3.7, we can bring one of these internal points across a boundary line by a $\Delta _k$-move, which reduces the number of internal points by one and allows us to apply the first inductive hypothesis.

Next, suppose the claim holds for all configurations Q with $|\text {int}(Q)| = |\text {int}(P)|$ and $|\text {nsnb}(Q)| < |\text {nsnb}(P)|$. By Lemma 3.9, we know that there is an internal point z in P such that one side of the region $R_z^\text {ex}$ is contained in a non-separating line in $\mathcal {A}^{\text {ex}}$. This means that while z may not be adjacent to a non-separating line, there is a finite sequence of moves across lines with at least one internal endpoint which takes z to a region which is adjacent to a non-separating line in $\mathcal {A}^{\text {ex}}$. Each move across lines with an internal endpoint preserves Property $\Delta _k$ by Lemma 3.8, after which we can perform a $\Delta _k$-move across the non-separating line by Lemma 3.7. This new configuration has fewer non-separating non-boundary lines, so by the second inductive hypothesis, it can be further transformed via $\Delta _k$-moves into a convex configuration and the proof is complete. $\square$

From a topological perspective, Theorem 3.10 can be interpreted as a statement about the configuration space of n points in the plane. To do so, recall that the weak Property $\Delta _k$ is a slight weakening of Property $\Delta _k$ which allows for one instance of three collinear points.

Corollary 3.11

(Theorem C) Let $k\in \{1,2\}$. The set of all configurations which satisfy the weak Property $\Delta _k$ forms a connected subspace of the configuration space of n points in $\mathbb {C}$.

Applying a move to a configuration P will certainly affect the noncrossing partition lattice $\textsc {NC}(P)$, and possibly even its isomorphism type. To close this section, we introduce a natural map between the larger partition lattices.

Fig. 10

The block-switching map ${\textsc {BS}}_m$, compared to the induced map $m_*$ for a fixed move m. In this example, ${\textsc {BS}}_m$ takes an element of $\textsc {NC}(P)$ to an element of $\textsc {NC}(m(P))$

Definition 3.12

Let $m:P \rightarrow m(P)$ be a move which brings the point $z\in P$ across the line $\ell$ in $\mathcal {A}^z$ and let $w_1$ and $w_2$ be the two points of P on $\ell$. The block-switching map ${\textsc {BS}}_m:\Pi (P) \rightarrow \Pi (m(P))$ is defined for each $\pi \in \Pi (P)$ as follows: if $w_1$ and $w_2$ share a block in $\pi$ and z belongs to a different block, then define ${\textsc {BS}}_m(\pi )$ to be the result of removing $\{w_1,w_2\}$ and $\{m(z)\}$ from their respective blocks in $m_*(\pi )$ and swapping them; otherwise, define ${\textsc {BS}}_m(\pi ) = m_*(\pi )$. See Fig. 10 for an illustration. Note that ${\textsc {BS}}_m$ is a rank-preserving bijection, but not an isomorphism.

4 Skewers

In this section, we introduce the notion of “skewering” in the line arrangement $\mathcal {A}$ and prove some technical results which come from Property $\Delta _2$. These tools are used in this section to prove that $\textsc {NC}(P)$ is graded when P satisfies Property $\Delta _2$ (Proposition 4.4) and in Sect. 6 to prove Theorem B. Throughout the rest of this section, let P be a configuration of n points in $\mathbb {C}$ and let $\mathcal {A}$ be the corresponding line arrangement.

Definition 4.1

For each pair of distinct lines $\ell _1,\ell _2 \in \mathcal {A}$, the intersection $\ell _1\cap \ell _2$ can be classified into one of four different types (without loss of generality):

• if $\ell _1\cap \ell _2 = \emptyset$, then $\ell _1$ and $\ell _2$ are parallel;

• if $\ell _1\cap \ell _2$ lies in $c(\ell _1)\cap c(\ell _2)$, then $\ell _1$ and $\ell _2$ intersect internally;

• if $\ell _1\cap \ell _2$ lies in neither $c(\ell _1)$ nor $c(\ell _2)$, then $\ell _1$ and $\ell _2$ intersect externally;

• if $\ell _1\cap \ell _2$ lies in $c(\ell _2)$ but not $c(\ell _1)$, then $\ell _1$ skewers $\ell _2$.

As a useful shorthand, we write $\ell _1\dashv \ell _2$ to mean that $\ell _1$ skewers $\ell _2$. When this is the case, note that the convex hull $c(\ell _1,\ell _2)$ contains one of the endpoints of $\ell _1$ as an internal point—we refer to this as the link vertex for the skewer. Finally, a skewering sequence is a collection of lines $\ell _1,\ldots ,\ell _k$ in $\mathcal {A}$ with $\ell _1 \dashv \ell _2 \dashv \cdots \dashv \ell _k$.

Fig. 11

A skewering tree for a configuration of 14 points which satisfies Property $\Delta _2$, but not Property $\Delta _1$. The tree consists of two skewering sequences: $\ell _0 \dashv \alpha _1 \dashv \alpha _2$ and $\ell _0 \dashv \beta _1$. For the sake of visual clarity, only the cores of the four lines have been drawn

Definition 4.2

Suppose that P satisfies Property $\Delta _2$. A skewering tree is a subset $T \subset \mathcal {A}$, together with two additional pieces of data—a special element $\ell _0 \in T$ called the initial line and a chosen closed half-plane bounded by $\ell _0$, which we call the positive side of $\ell _0$—with the following properties:

• no two elements of T intersect internally;

• for all $\ell \in T$ with $\ell \ne \ell _0$, there is a unique line $\ell ' \in T$ which skewers $\ell$;

• no element of P is the link vertex for more than one skewer.

We can also build a skewering tree inductively as follows: begin with an initial line $\ell _0\in \mathcal {A}$, select one of the two half-planes bounded by $\ell _0$ to be the positive side, and define $T = \{\ell _0\}$. Next, either stop here or add a line from $\mathcal {A}$ to T which is skewered by another element of T such that the requirements above remain satisfied. Repeat this process and stop at any point; the resulting set T is a skewering tree. See Fig. 11 for an example. In either construction, we refer to the non-initial lines in T which do not skewer any other lines as leaves. Finally, we say that a skewering tree is maximal if it is not properly contained in any other skewering tree in $\mathcal {A}$.

Fig. 12

If $\ell$ intersects the convex hull of $\ell _1$ and $\ell _2$ (denoted in blue), then the set of all endpoints has half of its elements in the interior (shown here as unfilled red dots)

The following lemma places fairly strong restrictions on the planar structure of skewering trees in configurations which satisfy Property $\Delta _2$.

Lemma 4.3

Suppose P satisfies Property $\Delta _2$, let $T \subset \mathcal {A}$ be a skewering tree, and let $\ell$, $\ell _1$, and $\ell _2$ be distinct lines in T such that $\ell$ is a leaf and $\ell _1 \dashv \ell _2$. Then $\ell$ does not intersect the convex hull $c(\ell _1,\ell _2)$.

Proof

First, suppose that $\ell$ is the last element in a skewering sequence which includes $\ell _1$ and $\ell _2$, i.e. that there is a skewering sequence $\ell _0 \dashv \alpha _1 \dashv \cdots \alpha _k$ such that $\ell = \alpha _k$, and $\alpha _i = \ell _1$ and $\alpha _{i+1} = \ell _2$ for some i. The set $V(\{\ell _0,\alpha _1,\ldots ,\alpha _k\})$ then consists of $2(k+1)$ points, of which at least k are internal: one for each non-leaf element in the skewering sequence. If $\ell$ were to intersect the convex hull $c(\ell _1,\ell _2)$, then one of the endpoints of $\ell$ would lie in the interior of the convex hull $c(\ell _1,\ell _2,\ell )$, but that would mean that the $2(k+1)$-element set described above has $k+1$ internal points, which contradicts our assumption that P satisfies Property $\Delta _2$.

On the other hand, suppose that $\ell$ is in a different skewering sequence than $\ell _1$ and $\ell _2$. That is, suppose that T contains the sequences $\ell _0 \dashv \alpha _1 \dashv \cdots \dashv \alpha _k$ and $\ell _0 \dashv \beta _1 \dashv \cdots \dashv \beta _m$, where $\alpha _k = \ell$, and $\beta _i = \ell _1$ and $\beta _{i+1} = \ell _2$ for some i. Then $V(\{\ell _0,\alpha _1,\ldots ,\alpha _k,\beta _1,\ldots ,\beta _m\})$ is a set of $2(k+m+1)$ points in P, of which at least $k+m$ are internal. Once again, if $\ell$ intersected the convex hull $c(\ell _1,\ell _2)$, this would imply that our set of $2(k+m+1)$ points has $k+m+1$ internal points, which again would violate Property $\Delta _2$. In both cases, we see that $\ell$ does not intersect $c(\ell _1,\ell _2)$. $\square$

Note that one may “prune” a skewering tree by iteratively removing leaves, and the result remains a skewering tree at each step. Thus, Lemma 4.3 further shows that the non-leaf elements of a skewering tree are similarly constrained.

As mentioned in Sect. 2, the full partition lattice $\Pi (P)$ is graded by the rank function $\rho (\pi ) = n-bl(\pi )$ for any choice of configuration P, but it is possible to construct P such that $\textsc {NC}(P)$ is not graded. For example, Fig. 13 depicts a noncrossing partition with three blocks which is covered by the maximum element ${\hat{1}}$ in $\textsc {NC}(P)$, but not in $\Pi (P)$. In the following proposition, we use Lemma 4.3 to show that configurations with Property $\Delta _2$ avoid this sort of behavior.

Fig. 13

No two blocks of this noncrossing partition can be combined to make a coarser noncrossing partition without also including the third, so this is a partition with three blocks which is covered by the maximum element ${\hat{1}}$, which has one block

Proposition 4.4

If P satisfies Property $\Delta _2$, then $\textsc {NC}(P)$ is graded.

Proof

Suppose P has Property $\Delta _2$ and let $\pi < \pi '$ in $\textsc {NC}(P)$ be a covering relation, i.e. there is no $\pi '' \in \textsc {NC}(P)$ with $\pi< \pi '' < \pi '$. Then $\pi '$ is obtained from $\pi$ in the following manner: there is a collection $\mathcal {B}$ of $k \ge 2$ blocks in $\pi$ which are removed and replaced by the union $U = \cup \mathcal {B}$ to create $\pi '$. If we can show that $k = 2$, then we have that $\rho (\pi ) + 1 = \rho (\pi ')$, and therefore $\rho$ is a rank function for $\textsc {NC}(P)$.

To this end, let $\mathcal {A}_{\pi }$ denote the lines in $\mathcal {A}$ which contain a side of $\textsc {Conv}(A)$ for some block $A \in \pi$ and let $A_1 \in \mathcal {B}$. Since $k\ge 2$, there is a side of $\textsc {Conv}(A_1)$ which does not lie in the boundary of $\textsc {Conv}(U)$; let $\ell _1 \in \mathcal {A}_\pi$ be the line which contains this side. If $\ell _1$ does not skewer any other line in $\mathcal {A}_\pi$, then each block in $\pi$ lies on one side of $\ell _1$ or the other, and therefore one may define a new partition $\pi ''$ by replacing $\mathcal {B}$ with two blocks, each of which is the union of all blocks in $\mathcal {B}$ with interior on a particular side of $\ell _1$. In this case, we would have $\pi \le \pi '' < \pi '$, which implies $\pi = \pi ''$ by our initial assumptions, thus $k=2$ and we are done.

Suppose $\ell _1$ does skewer another line $\ell _2 \in \mathcal {A}_\pi$. Without loss of generality, we may choose $\ell _2$ to be one of the “closest” to $\ell _1$ in the sense that the segment of $\ell _1$ between the intersection $\ell _1 \cap \ell _2$ and the core $c(\ell _1)$ does not intersect any elements of $\mathcal {A}_\pi$. Then $\ell _2$ does not contain a side of $\textsc {Conv}(U)$, so we may repeat the analysis in the previous paragraph: if $\ell _2$ does not skewer any other line in $\mathcal {A}_\pi$, then we are done. Otherwise, we can continue this process to obtain a skewering sequence $\ell _1 \dashv \ell _2 \dashv \cdots \dashv \ell _m$ such that each $\ell _i$ is an element of $\mathcal {A}_\pi$ which does not contain a side of $\textsc {Conv}(U)$, and by Lemma 4.3, we can guarantee that this sequence terminates in a line $\ell _m$ which does not skewer any line in $\mathcal {A}_\pi$. Applying the argument above to $\ell _m$, the proof is complete. $\square$

Next, we investigate how skewering trees decompose configurations into regions.

Definition 4.5

The union of all lines in a skewering tree T forms a subset of the plane with the structure of an unbounded graph, consisting of vertices, line segments, rays, and lines (one may equivalently view this as a graph embedding on the 2-dimensional sphere, viewed as the stereographic projection of the plane); let $\Gamma _T$ denote the unbounded graph obtained from this by removing any ray which does not contain the core of its corresponding line in T. We refer to $\Gamma _T$ as the planar realization of T, observing that $\Gamma _T$ is an acyclic graph, i.e. a tree. Note that while a pair of rays in $\Gamma _T$ might overlap, Lemma 4.3 implies that no ray in $\Gamma _T$ has a transverse intersection with a line segment.

Fig. 14

Cells associated to the skewering tree in Fig. 11. In this case, the positive side of the initial line $\ell _0$ is chosen to be the upper-right side; the corresponding cell is shaded orange

Definition 4.6

For each non-initial line $\ell$ in T, the core $c(\ell )$ is contained in the boundary for exactly one region of the complement $\mathbb {C}-\Gamma _T$ (since the other side of $\ell$ contains the core of the line which skewers it). The cell associated to $\ell$, denoted $C_\ell$, is the union of the interior of this region together with the line segment or ray which contains $c(\ell )$. Note that $C_\ell$ is a convex (possibly unbounded) subset of the plane which contains exactly one side of its boundary. The initial line $\ell _0$ bounds two regions and thus corresponds to two cells: $C^{+}_{\ell _0}$, which contains the core $c(\ell _0)$ and belongs to the positive side of $\ell _0$, and $C^-_{\ell _0}$, which does not. See Fig. 14 for an illustration. For any cell $C^{\pm }_\ell$, we write $V(C^{\pm }_\ell )$ to mean the intersection of P with $C^{\pm }_\ell$.

Lemma 4.7

The cells of a skewering tree are pairwise disjoint.

Proof

The interior of a cell for a skewering tree is a connected component of the complement $\mathbb {C}-\Gamma _T$, so it follows that the interiors of two cells overlap if and only if the interiors are identical. What remains to be shown is that no connected component of $\mathbb {C}-\Gamma _T$ belongs to the cores of two different lines in T.

First, we consider lines $\alpha _i$ and $\alpha _j$ which come from the same skewering sequence $\alpha _0 \dashv \alpha _1 \dashv \cdots \dashv \alpha _k$, where $0\le i < j \le k$. Note that the union

$$\begin{aligned} c(\alpha _i,\alpha _{i+1}) \cup c(\alpha _{i+1},\alpha _{i+2}) \cup \cdots \cup c(\alpha _{j-2},\alpha _{j-1}) \end{aligned}$$

is a connected subset of the plane and by Lemma 4.3, it cannot intersect $\alpha _j$. Thus, this subset lies on one side of $\alpha _j$ (the side which contains the cell $C_{\alpha _i}$), while the cell $C_{\alpha _j}$ lies on the other. Thus, the cells associated to $\alpha _i$ and $\alpha _j$ are disjoint.

Now, we consider lines $\alpha _i$ and $\beta _j$ which come from distinct skewering sequences $\alpha _0 \dashv \alpha _1 \dashv \cdots \dashv \alpha _k$ and $\beta _0 \dashv \beta _1 \dashv \cdots \dashv \beta _m$, where $\alpha _0 = \ell _0 = \beta _0$ and $i,j \ge 1$. Similar to the previous case, we observe that

$$\begin{aligned} \left( \bigcup _{t=1}^{i-1} c(\alpha _{t-1},\alpha _t)\right) \cup \left( \bigcup _{t=1}^{j-1} c(\beta _{t-1},\beta _t)\right) \end{aligned}$$

is a connected subset of the plane and by Lemma 4.3, it must be wholly contained in one of the four components of the complement $\mathbb {C}- (\alpha _i\cup \beta _j)$. In particular, it must belong to the unique component which contains the core of $\alpha _i$ and the core of $\beta _j$ in its boundary. Thus, this region of $\mathbb {C}- (\alpha _i\cup \beta _j)$ does not contain the cells for either $\alpha _i$ or $\beta _j$, which means that the two cells are disjoint - see Fig. 15 for an illustration. $\square$

Fig. 15

In the complement of the lines $\alpha _i$ and $\beta _j$, the region which has the cores of $\alpha _i$ and $\beta _j$ in its boundary (shaded blue here) must contain the cores of lines which skewer $\alpha _i$ and $\beta _j$, which means that the cells $C_{\alpha _i}$ and $C_{\beta _j}$ lie outside the shaded region

Lemma 4.8

Let P be a configuration of n points which satisfies Property $\Delta _2$. Then the cells of a skewering tree cover the elements of P.

Proof

Let $z\in P$, let T be a skewering tree and let $\Omega$ be the connected component of $\mathbb {C}- \Gamma _T$ which contains z. Suppose for the sake of contradiction that $\Omega$ does not belong to a cell of T. This immediately rules out the possibility that $\Omega$ touches only a single line in T, since that line would necessarily be a leaf and thus $\Omega$ would belong to the cell associated to that line. The remaining case to consider is that that there are distinct lines $\ell _1, \ell _2 \in T$ such that

• (1) $\ell _1$ and $\ell _2$ bound adjacent sides of $\Omega$, and

• (2) $\Omega$ lies in the unique component of $\mathbb {C}- (\ell _1\cup \ell _2)$ which has neither $c(\ell _1)$ nor $c(\ell _2)$ in its boundary.

Notice that this implies that the endpoints of $\ell _1$, the endpoints of $\ell _2$, and z form a 5-element subset of P where $\ell _i$ and $\ell _j$ each have one (non-link) endpoint in the interior of the convex hull. Similar to the proof of Lemma 4.7, we consider two cases according to whether $\ell _1$ and $\ell _2$ belong to the same skewering sequence or not.

First, suppose T contains a skewering sequence $\alpha _0 \dashv \alpha _1 \dashv \cdots \dashv \alpha _k$ such that $\ell _1 = \alpha _i$ and $\ell _2 = \alpha _j$, where $0\le i < j \le k$ and let A be the set containing both z and the endpoints of $\alpha _i, \alpha _{i+1},\ldots , \alpha _j$. Then $|A| = 2(j-i+1) + 1$ and we know that $j-i$ points in A are link vertices for skewers involving other lines in A. Together with the two points mentioned above, we have that A is a set of $2(j-i+1) + 1$ points where $j-i+2$ of them are internal, which contradicts our assumption that P satisfies Property $\Delta _2$.

Next, suppose that T contains skewering sequences $\alpha _0 \dashv \alpha _1 \dashv \cdots \dashv \alpha _k$ and $\beta _0 \dashv \beta _1 \dashv \cdots \dashv \beta _m$ such that $\ell _1 = \alpha _i$ and $\ell _2 = \beta _j$ for some $i,j \ge 1$, and let B be the set containing z together with the endpoints of $\ell _0,\alpha _1,\ldots ,\alpha _i,\beta _1,\ldots ,\beta _j$. Then $|B| = 2(i+j+1) + 1$, and at least $i+j$ elements of B are internal due to being link vertices of skewers in this set. By the same reasoning as in the previous case, we know that $i+j+2$ of the $2i+2j+3$ points in B are internal, which again contradicts our assumption that P satisfies Property $\Delta _2$.

Since both cases lead to a contradiction of Property $\Delta _2$, we conclude that z must in fact belong to a cell of the skewering tree T, and this completes the proof. $\square$

We now conclude this section by defining a subposet of $\textsc {NC}(P)$ associated to each skewering tree.

Definition 4.9

Let P be a configuration of n points satisfying Property $\Delta _2$ and let T be a skewering tree for P. We define the skewering interval $\textsc {NC}(P,T)$ to be the subposet of $\textsc {NC}(P)$ consisting of all partitions $\pi$ satisfying the following conditions:

• (1) for each $\ell \in T$, the endpoints of $\ell$ share a block in $\pi$;

• (2) endpoints of distinct lines in T belong to distinct blocks in $\pi$;

• (3) each block in $\pi$ has a convex hull which lies in a cell of T.

As the name suggests, $\textsc {NC}(P,T)$ is an interval in $\textsc {NC}(P)$. Let ${\hat{0}}_T$ be the partition in $\textsc {NC}(P)$ for which the only non-singleton blocks are $V(\ell )$ for each $\ell \in T$. Let ${\hat{1}}_T$ be the partition where two points in P belong to the same block if and only if they belong to the same cell of T (note that this requires us to choose a positive side for $\ell _0$ before discussing the skewering interval). Then $\textsc {NC}(P,T)$ is the interval $[{\hat{0}}_T,{\hat{1}}_T]$. See Fig. 16.

We close the section with some features of skewering intervals which will be useful in the proof of Theorem B. To begin, we provide a product decomposition for skewering intervals.

Definition 4.10

Let $\ell$ be a boundary line in the arrangement $\mathcal {A}$ corresponding to P. Define $\pi _\ell$ to be the partition of P in which the two endpoints of $\ell$ share a block, while every other block is a singleton. Further, let $\textsc {NC}_\ell (P)$ denote the interval $[\pi _\ell ,{\hat{1}}]$ in $\textsc {NC}(P)$.

Fig. 16

The minimum and maximum elements for the interval associated to the skewering tree drawn in Fig. 11

Lemma 4.11

Let T be a skewering tree for P. Then the skewering interval $\textsc {NC}(P,T)$ is isomorphic to the product

$$\begin{aligned} \left( \prod _{\begin{array}{c} \ell \in T\\ \ell \ne \ell _0 \end{array}} \textsc {NC}_\ell (V(C_\ell ))\right) \times \textsc {NC}_{\ell _0}(V(C_{\ell _0}^+)) \times \textsc {NC}(V(C_{\ell _0}^-)). \end{aligned}$$

Proof

This follows immediately from Definitions 4.9 and 4.10. $\square$

Next, we note that each skewering interval is “centered” in the sense that the rank of its minimum element is equal to the corank of its maximum.

Definition 4.12

Let $\pi _0$ and $\pi _1$ be elements of $\textsc {NC}(P)$ with $\pi _0 \le \pi _1$. We say that the interval $[\pi _0,\pi _1]$ is centered if $\rho (\pi _0)+\rho (\pi _1) = |P| - 1$, or equivalently if $bl(\pi _0) + bl(\pi _1) = |P| + 1$.

Lemma 4.13

Each skewering interval $\textsc {NC}(P,T)$ is a centered subposet of $\textsc {NC}(P)$.

Proof

Let T be a skewering tree for P. Then the partition ${\hat{0}}_T$ has rank |T| and ${\hat{1}}_T$ has rank $|P| - (|T|+1)$, so the interval $[{\hat{0}}_T,{\hat{1}}_T]$ is centered. $\square$

Finally, we examine the ways in which skewering intervals can intersect.

Lemma 4.14

If P has Property $\Delta _2$, then distinct maximal skewering trees with the same initial line and choice of positive side yield disjoint skewering intervals.

Proof

Let T and $T'$ be distinct maximal skewering trees and suppose that the partition $\pi$ is contained in both $\textsc {NC}(P,T)$ and $\textsc {NC}(P,T')$. In other words, both ${\hat{0}}_T \le \pi \le {\hat{1}}_T$ and ${\hat{0}}_{T'} \le \pi \le {\hat{1}}_{T'}$. Since T and $T'$ are distinct and maximal, there must be lines $\alpha ,\ell$, and $\ell '$ such that $\alpha ,\ell \in T$, $\alpha ,\ell ' \in T'$, and $\alpha$ skewers both $\ell$ and $\ell '$. Note that by Lemma 4.3, $\ell$ and $\ell '$ cannot skewer one another, and by definition of a skewering tree, $\ell$ and $\ell '$ do not intersect internally. Therefore the two lines are either parallel or have an external intersection.

Let $H^-_{\alpha ,\ell }$ and $H^-_{\alpha ,\ell '}$ denote the closed half-planes bounded by $\ell$ and $\ell '$ respectively which do not include the core of $\alpha$. Since $\alpha$ skewers $\ell$, we know that the cell $C_\ell$ (and therefore the core $c(\ell )$) must belong to $H^-_{\alpha ,\ell }$; the analogous statement holds for $\ell '$. By combining the inequalities above, we see that ${\hat{0}}_T \le {\hat{1}}_{T'}$, which means that the core $c(\ell )$ must be contained in the cell $C_{\ell '}$, which implies that $c(\ell )$ belongs to $H^-_{\alpha ,\ell '}$. Similarly, the fact that ${\hat{0}}_{T'} \le {\hat{1}}_T$ tells us that $c(\ell ')$ is contained in $H^-_{\alpha ,\ell }$.

Combining all of the above, we know that the intersection $H^-_{\alpha ,\ell }\cap H^-_{\alpha ,\ell '}$ includes both $c(\ell )$ and $c(\ell ')$, but not $c(\alpha )$. However, by the same reasoning used in the proof of Lemma 4.7, this is precisely the region in which the core of $\alpha$ must be placed for it to skewer both $\ell$ and $\ell '$. Therefore, we have a contradiction, so the skewering intervals for T and $T'$ must be disjoint. $\square$

5 Property $\Delta _1$ and Catalan Numbers

Our strategy for proving Theorem A is to show that if P satisfies Property $\Delta _1$, then applying a $\Delta _1$-move to P does not change the size of $\textsc {NC}(P)$. From here, applying Theorem 3.10 completes the proof. We begin with a useful lemma and some terminology, then give the proof of Theorem A.

Fig. 17

If the region $R_z$ has a side which belongs to a line through an interior point v, then z and v must both belong to a common triangle

Lemma 5.1

Suppose P satisfies Property $\Delta _1$ and let $z\in \text {int}(P)$. Then $R_z = R_z^\text {ex}$.

Proof

We know by definition that $R_z$ and $R_z^\text {ex}$ are convex polygons with $R_z \subseteq R_z^\text {ex}$, so we just need to show that each of its sides is a subset of a line in $\mathcal {A}^{\text {ex}}$. Suppose that one of the sides for $R_z$ is a subset of a line $\ell$ with endpoints u and v, where v (and possibly u as well) is internal. Then $\ell$ must not contain any other points in P (since P is assumed to be in general position), so it must eventually intersect the boundary of $\textsc {Conv}(P)$ in some edge between extremal vertices $w_1$ and $w_2$, and thus v lies within the triangle with vertex set $\{u,w_1,w_2\}$. Since $\ell$ was assumed to contain a side of $R_z$, we know that z must also be contained in the same triangle—see Fig. 17 for an illustration. But this implies that P does not satisfy Property $\Delta _1$, which is a contradiction. $\square$

Remark 5.2

Let P be a configuration satisfying Property $\Delta _1$, suppose that $z\in P$ is adjacent to a line $\ell$ in $\mathcal {A}^{\text {ex}}$, and let $\pi$ be a partition of P. Since P satisfies Property $\Delta _1$, we know by Lemma 5.1 that $\ell$ contains an edge of the region $R_z$. In other words, there are no lines in the arrangement $\mathcal {A}^z$ which lie between $\ell$ and z. This implies that if $B_z$ and $B_\ell$ are the blocks in $\pi$ containing z and the endpoints of $\ell$ respectively, then $\textsc {Conv}(B_z)$ and $\textsc {Conv}(B_\ell )$ are disjoint if and only if $B_z \cap H_{z,\ell }^- = \emptyset$ and $B_\ell \cap H_{z,\ell }^+ = \emptyset$.

Definition 5.3

Let $m:P \rightarrow m(P)$ be a move. We say that $\pi \in \Pi (P)$ is pre-m-noncrossing if $\pi$ is noncrossing, but its image $m_*(\pi ) \in \Pi (m(P))$ is not. Similarly, we say that $\mu \in \Pi (m(P))$ is post-m-noncrossing if $\mu$ is noncrossing, but its preimage $m_*^{-1}(\mu ) \in \Pi (P)$ is not. Then $\textsc {NC}(m(P))$ can be obtained from $\textsc {NC}(P)$ by the following procedure: remove partitions which are pre-m-noncrossing, apply $m_*$ to all remaining elements, then add in the post-m-noncrossing partitions.

We are now ready to prove the main theorem of this section.

Theorem 5.4

(Theorem A) Let $P \subset \mathbb {C}$ be a configuration of n points which satisfies Property $\Delta _1$. Then $\textsc {NC}(P)$ is a rank-symmetric graded lattice, and the number of elements with rank k is the Narayana number ${N}_{n,k}$. In particular, $|\textsc {NC}(P)| = C_{n} = |{\textsc {NC}}_{n}|$.

Proof

By Theorem 3.10, we know there is a sequence of $\Delta _1$-moves which transforms P into a convex configuration, for which we know the lattice of noncrossing partitions is isomorphic to ${\textsc {NC}}_{n}$. Therefore, we need only show that if $m:P\rightarrow m(P)$ is a $\Delta _1$-move, then the lattices $\textsc {NC}(P)$ and $\textsc {NC}(m(P))$ have the same number of elements in each rank.

Our strategy is to show that the block-switching map ${\textsc {BS}}_{m}$ restricts to a rank-preserving bijection $\textsc {NC}(P) \rightarrow \textsc {NC}(m(P))$. Suppose that $m:P\rightarrow m(P)$ is a $\Delta _1$-move which brings the point z across the line $\ell$ in $\mathcal {A}^{\text {ex}}$, where the endpoints of $\ell$ are $w_1$ and $w_2$, and let $m(z) = y$.

Let $\pi$ be a partition of P. Applying Remark 5.2 and the fact that $H_{y,\ell }^- = H_{z,\ell }^+$ and $H_{y,\ell }^+ = H_{z,\ell }^-$, we have the following sequence of equivalences:

$$\begin{aligned} \pi \text { is pre-} m \text {-noncrossing}&\leftrightarrow \begin{aligned}&\text {in } \pi {:}\, B_z \cap H_{z,\ell }^- \text { and } B_\ell \cap H_{z,\ell }^+ \text { are empty;} \\&\text {either } B_{y} \cap H_{y,\ell }^- \text { or } B_\ell \cap H_{y,\ell }^+ \text { is nonempty} \end{aligned} \\&\leftrightarrow \begin{aligned}&\text {in } {\textsc {BS}}_m(\pi ){:} \, B_\ell \cap H_{z,\ell }^- \text { and } B_{y} \cap H_{z,\ell }^+ \text { are empty;} \\&\text {either }B_{\ell } \cap H_{y,\ell }^- \text { or } B_{z} \cap H_{y,\ell }^+ \text { is nonempty} \end{aligned} \\&\leftrightarrow \begin{aligned}&\text {in } {\textsc {BS}}_m(\pi ){:}\, B_\ell \cap H_{y,\ell }^+ \text { and } B_{y} \cap H_{y,\ell }^- \text { are empty;} \\&\text {either } B_{\ell } \cap H_{z,\ell }^+ \text { or } B_{z} \cap H_{z,\ell }^- \text { is nonempty} \end{aligned} \\&\leftrightarrow {\textsc {BS}}_m(\pi ) \text { is post-} m \text {-noncrossing} \end{aligned}$$

Thus, the block-switching map ${\textsc {BS}}_m$ induces a rank-preserving bijection between the pre-m-noncrossing partitions of P and the post-m-noncrossing partitions of m(P), and we are done. $\square$

6 Property $\Delta _2$ and Rank Symmetry

Fig. 18

An atom in $\textsc {NC}(P)$ with the corresponding coatom

We now turn our attention to Theorem B, in which we demonstrate that if P satisfies Property $\Delta _2$, then $\textsc {NC}(P)$ is rank-symmetric. As a first step, one could write an explicit bijection between the atoms and coatoms of $\textsc {NC}(P)$, where P is an arbitrary configuration of n points. This was previously demonstrated by Razen and Welzl in the special case where P is in general position [6] and is straightforward to generalize. In short: each atom is determined by a single line in $\mathcal {A}$, and by slightly rotating this line counterclockwise about the midpoint of its core, we obtain a line which divides P into two pieces, thus producing a coatom in $\textsc {NC}(P)$—see Fig. 18 for an illustration.

One could hypothetically prove Theorem B by extending the map above to a rank-reversing bijection from $\textsc {NC}(P)$ to itself, but this seems intractable in general. Instead, our proof technique is similar to that of Theorem A, although the weakening of our hypotheses from Property $\Delta _1$ to Property $\Delta _2$ means that Lemma 5.1 no longer holds. That is, it is possible that while some points in P are adjacent to non-separating lines in $\mathcal {A}$, none of these lines are in $\mathcal {A}^{\text {ex}}$. As a result, we must account for moves which take an interior point across a line with an interior endpoint, which in turn means we need to understand situations where a partition block overlaps with the line being moved across. To do so, we use the results on skewers developed in Sect. 4.

First, we require a technical lemma regarding the interval ${\textsc {NC}}_\ell (P)$ introduced in Definition 4.10. For the remainder of this section, let P denote a configuration of n points in $\mathbb {C}$ which satisfies Property $\Delta _2$.

Definition 6.1

Let $\ell$ be a boundary line in the arrangement $\mathcal {A}$ corresponding to P. We say that P satisfies Property $\Delta _1$ relative to $\ell$ if each $B\subseteq P$ with $V(\ell ) \subseteq B$ has at most $\lfloor \frac{|B|-2}{2} \rfloor$ internal points.

The motivation for the preceding definition comes from its appearance in certain skewering trees for configurations which satisfy Property $\Delta _2$.

Lemma 6.2

Let T be a skewering tree for P with initial line $\ell _0$. Suppose that $\ell _1$ is a line in T with $\ell _0 \dashv \ell _1$, and that y is a point in P which lies in the convex hull $c(\ell _0,\ell _1)$ on the non-positive side of $\ell _0$. Then $V(C_{\ell _0}^+)$ satisfies Property $\Delta _1$ relative to $\ell _0$, and for each $\ell \in T$ with $\ell \ne \ell _0$, $V(C_\ell )$ satisfies Property $\Delta _1$ relative to $\ell$.

Proof

We prove both claims by contradiction. To start, suppose that $V(C_{\ell _0}^+)$ does not satisfy Property $\Delta _1$ relative to $\ell _0$. Then there is a subset $B\subseteq V(C_{\ell _0}^+)$ which contains the endpoints of $\ell _0$ such that B has more than $\lfloor \frac{|B|-2}{2} \rfloor$ internal points. If we define $B' = B\cup V(\ell _1) \cup \{y\}$, then the internal points of B, together with y and one endpoint of $\ell _0$, are all internal points of $B'$. Therefore, $B'$ has more than

$$\begin{aligned} \left\lfloor \frac{|B| - 2}{2}\right\rfloor + 2 = \left\lfloor \frac{|B| + 2}{2}\right\rfloor = \left\lfloor \frac{|B'| - 1}{2} \right\rfloor \end{aligned}$$

internal points, which violates the assumption that P satisfies Property $\Delta _2$. Thus $V(C_{\ell _0}^+)$ satisfies Property $\Delta _1$ relative to $\ell _0$.

Similarly, let $\ell \in T$ with $\ell \ne \ell _0$ and suppose that there is a subset $B\subseteq V(C_\ell )$ which contains the endpoints of $\ell$ such that B has more than $\lfloor \frac{|B|-2}{2} \rfloor$ internal points. If $\ell = \ell _1$, then we can define $B' = B\cup V(\ell _0) \cup \{y\}$ and observe by the same reasoning as above that $B'$ has more than $\lfloor \frac{|B'|-1}{2} \rfloor$ internal points, which provokes a contradiction. Suppose instead that $\ell \ne \ell _1$. Then there is a skewering sequence $\alpha _1 \dashv \cdots \dashv \alpha _k \dashv \ell$ such that either $\ell _0 \dashv \alpha _1$ or $\ell _0 \dashv \ell _1 \dashv \alpha _1$. In either case, define

$$\begin{aligned} B' = B\cup V(\ell _0) \cup V(\ell _1) \cup \{y\} \cup V(\alpha _1) \cup \cdots \cup V(\alpha _k) \end{aligned}$$

and observe that the internal points of $B'$ include all the internal points of B, as well as one endpoint of each $\alpha _i$, one endpoint of $\ell _0$, and y. Therefore, the number of internal points in $B'$ is more than

$$\begin{aligned} \left\lfloor \frac{|B|-2}{2} \right\rfloor + k + 3 = \left\lfloor \frac{|B|+2k+4}{2} \right\rfloor = \left\lfloor \frac{|B'|-1}{2} \right\rfloor , \end{aligned}$$

which violates Property $\Delta _2$. Thus, $V(C_\ell )$ must have Property $\Delta _1$ relative to $\ell$. $\square$

Lemma 6.3

Suppose that P satisfies Property $\Delta _1$ relative to the boundary line $\ell$ and that for each proper subset $Q\subset P$, the poset of noncrossing partitions $\textsc {NC}(Q)$ is rank-symmetric. Then ${\textsc {NC}}_\ell (P)$ is rank-symmetric.

Proof

We proceed by induction on the number of internal points of P. First, if P has no internal points, then since $\ell$ was assumed to be a boundary line, we can see that ${\textsc {NC}}_\ell (P)$ is isomorphic to ${\textsc {NC}}_{n-1}$, which is rank-symmetric. Now, suppose that the claim is true for any configuration with up to $k-1$ internal points which satisfies the lemma’s hypotheses, and let P be a configuration with k internal points such that for all proper subsets $Q\subset P$, we know that $\textsc {NC}(Q)$ is rank-symmetric. Let u and v be the endpoints of $\ell$, and let $P^v$ denote the complement $P - \{v\}$. We will compare the interval $[\pi _\ell ,{\hat{1}}]\subset \textsc {NC}(P)$ to the noncrossing partition lattice $\textsc {NC}(P^v)$, which we know is rank-symmetric by assumption.

Define the map $\phi _v :{\textsc {NC}}_\ell (P) \rightarrow \textsc {NC}(P^v)$ by removing v from each partition in the domain and observe that $\phi _v$ is always injective, but typically not surjective. Our goal is to show that ${\textsc {NC}}_\ell (P)$ is rank-symmetric; since $\textsc {NC}(P^v)$ is assumed to be rank-symmetric and $\phi _v$ is injective, it suffices to show that the complement $\textsc {NC}(P^v) - \phi _v({\textsc {NC}}_\ell (P))$ is a union of centered, disjoint, rank-symmetric intervals.

Let $W\subset P -\{u,v\}$ be the set of all points w with the property that $\text {int}(\{u,v,w\})$ is nonempty and note that since P satisfies Property $\Delta _1$ relative to $\ell$, we must have $|\text {int}(\{u,v,w\})| = 1$. Then $\textsc {NC}(P^v) - \phi _v({\textsc {NC}}_\ell (P))$ is the collection of all partitions $\sigma$ of $P^v$ with a block which contains both u and a point $w\in W$, but not the unique point in $\text {int}(\{u,v,w\})$. Rephrasing this characterization in the language of skewering trees, let $\mathcal {T}$ be the collection of skewering trees for $P^v$ such that the initial line $\ell _0$ has endpoints u and w for some $w\in W$ and the positive side of $\ell _0$ is chosen to be the one which does not include v; then a partition $\sigma \in \textsc {NC}(P^v)$ lies in $\textsc {NC}(P^v) - \phi _v({\textsc {NC}}_\ell (P))$ if and only if it belongs to the skewering interval for some skewering tree in $\mathcal {T}$. Together with Lemmas 4.13 and  4.14, this tells us that the skewering intervals for trees in $\mathcal {T}$ form a collection of centered and disjoint intervals whose union is $\textsc {NC}(P^v) - \phi _v({\textsc {NC}}_\ell (P))$. All that remains is to show that each such skewering interval is rank-symmetric.

Fix a skewering tree $T \in \mathcal {T}$ and consider the skewering interval $\textsc {NC}(P^v,T)$. By Lemma 4.11, we have a poset isomorphism

$$\begin{aligned} \textsc {NC}(P^v,T) \cong \left( \prod _{\begin{array}{c} \alpha \in T \\ \alpha \ne \ell _0 \end{array}} {\textsc {NC}}_\alpha (P^v \cap C_\alpha ) \right) \times {\textsc {NC}}_{\ell _0}(P^v \cap C^+_{\ell _0})\times \textsc {NC}(P^v \cap C^-_{\ell _0}). \end{aligned}$$

By our assumption that every proper subset of P has a rank-symmetric lattice of noncrossing partitions, we know that $\textsc {NC}(P^v\cap C^-_{\ell _0})$ is rank-symmetric. For the middle term in the product, we claim that the set $P^v \cap C^+_{\ell _0}$ satisfies Property $\Delta _1$ relative to the boundary line $\ell _0$. To see this, recall that the endpoints of $\ell _0$ are the extremal point u and some $w\in W$, and define $w'$ be the unique point of P which lies in the interior of $\textsc {Conv}(\{u,v,w\}$. If there were points $x, x', x'' \in P^v \cap C^+_{\ell _0}$ such that both $x'$ and $x''$ lie in the convex hull $\textsc {Conv}(\{u,w,x\})$, then $\{u,v,w,w',x,x',x''\}$ would be a set of seven points, at least three of which are internal ($w'$, $x'$, and $x''$). This would violate our assumption that P has Property $\Delta _1$ relative to $\ell$, so it must be the case that $P^v - C^+_{\ell _0}$ satisfies Property $\Delta _1$ relative to $\ell _0$, and by our inductive hypothesis, we know that ${\textsc {NC}}_{\ell _0}(P^v \cap C^+_{\ell _0})$ is rank-symmetric.

Generalizing this argument, we now show that $P^v \cap C_\alpha$ satisfies Property $\Delta _1$ relative to $\alpha$ for each non-initial line $\alpha$ in T. Let $\alpha _1 \dashv \cdots \dashv \alpha _m$ be a skewering sequence in T such that $\alpha _1 = \ell _0$ and $\alpha _m = \alpha$. If there are points $x,x',x'' \in P^v \cap C_\alpha$ such that $x'$ and $x''$ lie in the triangle formed by x and the endpoints of $\alpha$, and if w and $w'$ are defined as above, then

$$\begin{aligned} \{x,x',x'',w',v\} \cup V(\alpha _1) \cup \cdots \cup V(\alpha _m) \end{aligned}$$

is a set of $2m+5$ points, of which at least $m+2$ points must be internal—see Fig. 19 for an illustration. Since this set contains both u and v, this violates our assumption that P satisfies Property $\Delta _1$ relative to $\ell$, so we may conclude that $P^v \cap C_\alpha$ satisfies Property $\Delta _1$ relative to $\alpha$, as desired. By the inductive hypothesis, ${\textsc {NC}}_\alpha (P^v \cap C_\alpha )$ is rank-symmetric.

Fig. 19

If $\alpha _1\dashv \alpha _2\dashv \alpha _3$ is a skewering sequence for P and the points in the cell for $\alpha _3$ do not satisfy Property $\Delta _1$ relative to $\alpha _3$ (illustrated on the left in blue), then there is a set of points containing u and v (illustrated on right in red) which demonstrate that P does not satisfy Property $\Delta _1$ relative to the line with endpoints u and v

Finally, we have that $\textsc {NC}(P^v,T)$ is a product of rank-symmetric posets and is therefore rank-symmetric itself, which completes the proof. $\square$

Theorem 6.4

(Theorem B) Let $P \subset \mathbb {C}$ be a set of n distinct points which satisfies Property $\Delta _2$. Then $\textsc {NC}(P)$ is a rank-symmetric graded lattice.

Proof

We proceed by induction on the number of internal points for P. When P has no internal points, $\textsc {NC}(P)$ is isomorphic to the classical noncrossing partition lattice ${\textsc {NC}}_{n}$, which is rank-symmetric. Now, suppose that every configuration with fewer than $|\text {int}(P)|$ internal points has a rank-symmetric lattice of noncrossing partitions; we will show that $\textsc {NC}(P)$ is rank-symmetric as well. By Theorem 3.10, there is a sequence of $\Delta _2$-moves which transforms P into a convex configuration, for which the lattice of noncrossing partitions is isomorphic to ${\textsc {NC}}_{n}$ and thus rank-symmetric. The only remaining step is to prove that if $m:P \rightarrow m(P)$ is a $\Delta _2$-move, then $\textsc {NC}(P)$ is rank-symmetric if and only if $\textsc {NC}(m(P))$ is rank-symmetric.

In the proof of Theorem A, we showed that when P satisfies Property $\Delta _1$, the block-switching map ${\textsc {BS}}_m$ is a rank-preserving bijection between pre-m-noncrossing partitions of P and post-m-noncrossing partitions of m(P). If P is only assumed to satisfy Property $\Delta _2$, the block switching map might not be a bijection; there may be partitions $\pi \in \textsc {NC}(P)$ such that ${\textsc {BS}}_m(\pi )$ is not in $\textsc {NC}(m(P))$, or elements $\sigma \in \textsc {NC}(m(P))$ such that ${\textsc {BS}}_m^{-1}(\sigma )$ does not lie in $\textsc {NC}(P)$. To complete the proof, we must show that these two collections are rank-symmetric subposets of $\textsc {NC}(P)$ and $\textsc {NC}(m(P))$ respectively. By symmetry, it suffices to examine the first collection.

Suppose that the move m takes a point $z\in P$ across a line $\ell \in \mathcal {A}$ with endpoints $e(\ell ) = \{w_1,w_2\}$ and let $F_m(P)$ denote the noncrossing partitions of P which fail to be accounted for by the block-switching map ${\textsc {BS}}_m$; that is,

$$\begin{aligned} F_m(P) = \{\pi \in \textsc {NC}(P) \mid {\textsc {BS}}_m(\pi ) \not \in \textsc {NC}(m(P))\}. \end{aligned}$$

If $F_m(P)$ is empty, then there is nothing to prove. Otherwise, for each $\pi \in F_m(P)$, there are points $x,y\in P$ such that $y,z \in \text {int}(\{w_1,w_2,x\})$, and these five points are divided into three distinct blocks of $\pi$ (each of which might contain other points) as follows: x and z belong to one block, $w_1$ and $w_2$ belong to another, and y lies in a third—see Fig. 20. Using this characterization, we will prove that (1) $F_m(P)$ is a union of skewering intervals, (2) each of these skewering intervals is centered and rank-symmetric, and (3) intersections of the skewering intervals are centered and rank-symmetric.

Fig. 20

For the depicted partition $\pi$ of P, if m moves z across the line containing $w_1$ and $w_2$, then ${\textsc {BS}}_m(\pi )$ is not an element of $\textsc {NC}(m(P))$ since it has $w_1$, $w_2$, and x together in a single block without the interior point y

More concretely, let $y_1,\ldots ,y_k$ be the points in P such that for each $i\in \{1,\ldots ,k\}$, there is some $x_i\in P$ such that $w_1$, $w_2$, and $x_i$ form a triangle with interior points z and $y_i$ (and nothing else, since P satisfies Property $\Delta _2$). When there is more than one choice for $x_i$, we select the unique option where all other possible choices lie in the half-plane bounded by $x_i$ and z which does not contain $y_i$. Then $F_m(P)$ consists of all partitions in $\textsc {NC}(P)$ where there exists an i such that z and $x_i$ belong to the same block, $w_1$ and $w_2$ belong to a different block, and $y_i$ belongs to a third block.

Let $\ell _i$ denote the line with endpoints $x_i$ and z. Since z is adjacent to the line $\ell$, we know that $\ell _i$ must skewer $\ell$. The characterization above can thus be rephrased: the partition $\pi$ belongs to $F_m(P)$ if and only if $\pi$ lies in a skewering interval $\textsc {NC}(P,T_i)$ for some skewering tree $T_i$ with initial line $\ell _i$ such that $\ell \in T$ and $y_i$ is on the non-positive side of $\ell _i$. It is straightforward to see that each such skewering interval is a subset of $F_m(P)$, so we may conclude that $F_m(P)$ is a union of skewering intervals.

Each skewering interval in the union is centered by Lemma 4.13, which implies that $F_m(P)$ is itself a centered subposet of $\textsc {NC}(P)$. By Lemma 4.11, each skewering interval $\textsc {NC}(P,T_i)$ decomposes into a product of posets; the factor $\textsc {NC}(V(C_{\ell _i}^-))$ is rank-symmetric by our inductive hypothesis and terms of the form ${\textsc {NC}}_\ell (V(C_\ell ))$ and ${\textsc {NC}}_{\ell _i}(V(C_{\ell _i}^{+}))$ are rank-symmetric by Lemmas 6.2 and 6.3. Since the product of rank-symmetric posets is itself rank-symmetric, we may conclude that the interval $\textsc {NC}(P,T_i)$ is rank-symmetric.

Fig. 21

On the left and right, the minimum and maximum elements for a skewering interval are given. The intersection of these two intervals is another interval, albeit one which does not arise from a skewering tree; its minimum and maximum elements are displayed in the center

We are now at the final step: examining how the skewering intervals which make up $F_m(P)$ can intersect. First, note that if $T_i$ and $T_i'$ are two distinct skewering trees in the union which both use $\ell _i$ as the initial line, we know by Lemma 4.14 that $\textsc {NC}(P,T_i)\cap \textsc {NC}(P,T_i')$ is empty. Next, consider two skewering trees $T_i$ and $T_j$ which appear in the union with $i\ne j$. The intersection $\textsc {NC}(P,T_i)\cap \textsc {NC}(P,T_j)$ is nonempty precisely when the cells $C_{\ell _i}^+$ for $T_i$ and $C_{\ell _j}^+$ for $T_j$ have an intersection which excludes both $y_i$ and $y_j$ (note that this precludes the possibility of nonempty triple intersections for skewering intervals). When this is the case, the two skewering intervals intersect in the interval $[{\hat{0}}_{T_i} \vee {\hat{0}}_{T_j}, {\hat{1}}_{T_i} \wedge {\hat{1}}_{T_j}]$; this is not a skewering interval, but it has many of the associated properties. By similar arguments to those given in Lemmas 4.3, 4.7 and 4.8, we can see that the minimum element ${\hat{0}}_{T_i} \vee {\hat{0}}_{T_j}$ consists of an edge for each non-initial line in $T_i$ and $T_j$, together with the triangle with vertices $y_i$, $y_j$, and z. This triangle has three cells associated to it: one containing $z_i$, one containing $z_j$, and one which is the intersection of $C_{\ell _i}^+$ and $C_{\ell _j}^+$ (thus containing the triangle itself). Each other edge has a well-defined cell in the same way as a typical skewering tree does. The maximal element ${\hat{1}}_{T_i} \wedge {\hat{1}}_{T_j}$ is constructed using these cells in the same manner as for skewering trees—see Fig. 21 for an illustration. Putting this all together, $[{\hat{0}}_{T_i} \vee {\hat{0}}_{T_j}, {\hat{1}}_{T_i} \wedge {\hat{1}}_{T_j}]$ admits a decomposition similar to the one described in Lemma 4.11, so by Lemmas 6.2 and 6.3, this interval is centered and rank-symmetric.

In summary, we have shown that $F_m(P)$ is a union of centered and rank-symmetric intervals in $\textsc {NC}(P)$, whose pairwise intersections are themselves centered and rank-symmetric and whose k-wise intersections are empty when $k>2$. Therefore, $F_m(P)$ is centered and rank-symmetric, which completes the proof. $\square$