Crossing-Free Perfect Matchings in Wheel Point Sets

Abstract

Consider a planar finite point set P, no three points on a line and exactly one point not extreme in P. We call this a wheel set and we are interested in pm(P), the number of crossing-free perfect matchings on P. (If, contrary to our assumption, all points in a set S are extreme, i.e. in convex position, then it is well-known that pm(S) = C _m , the mth Catalan number, $m:= \frac{\vert S\vert } {2}$.)

We give exact tight upper and lower bounds on pm(P) depending on the cardinality of the wheel set P. Simplified to its asymptotics in terms of C _m , these yield

$$\displaystyle{ \frac{9} {8}C_{m}(1 + o(1)) \leq \mathsf{pm}(P) \leq \frac{3} {2}C_{m}(1 + o(1))\,m:= \frac{\vert P\vert } {2}. }$$

We characterize the wheel sets (order types) which maximize or minimize pm(P). Moreover, among all sets S of a given size not in convex position, pm(S) is minimized for some wheel set. Therefore, leaving convex position increases the number of crossing-free perfect matchings by at least a factor of $\frac{9} {8}$ (in the limit as | S | grows). We can also show that pm(P) can be computed efficiently.

A connection to origin embracing triangles is briefly discussed.

This research was initiated at the 13th Gremo’s Workshop on Open Problems (GWOP), Feldis, Switzerland, June 1–5, 2015. Research of the first author was partially supported by Swiss National Science Foundation grants 200021-137574 and 200020-144531. The second author acknowledges financial support from EuroCores/EuroGiga/ComPoSe, Swiss National Science Foundation grant 20GG21 134318/1.

Appendix: Catalan Facts

Fact A.1 (Definition & Asymptotics)

For all $m \in \mathbb{N}_{0}$,

$$\displaystyle{ C_{m}:= \frac{1} {m + 1}{2m\choose m} = \frac{(2m)!} {(m + 1)!m!} = \Theta \left ( \frac{1} {m^{3/2}}4^{m}\right )\ . }$$

Fact A.2 (Simple Recurrence)

For all $m \in \mathbb{N}_{0}$ ,

$$\displaystyle{ 2 \cdot \frac{2m - 1} {m + 1} \cdot C_{m-1} = C_{m} = \frac{1} {2} \cdot \frac{m + 2} {2m + 1} \cdot C_{m+1}\ . }$$

Fact A.3 (Segner Recurrence)

For all $m \in \mathbb{N}$ ,

$$\displaystyle{ C_{m} = C_{0}C_{m-1} + C_{1}C_{m-2} + \cdots \,C_{m-1}C_{0} =\sum _{ i=0}^{m-1}C_{ i}C_{m-1-i}. }$$

Fact A.4

For $k,\ell\in \mathbb{N}$ ,

$$\displaystyle{ C_{k}C_{\ell} < C_{k-1}C_{\ell+1}\mathit{\mbox{ iff }}k \leq \ell }$$

Proof

$$\displaystyle{ C_{k}C_{\ell} = 2\frac{2k - 1} {k + 1} C_{k-1} \cdot \frac{1} {2} \frac{\ell+2} {2\ell + 1}C_{\ell+1} = \frac{(2k - 1)(\ell+2)} {(k + 1)(2\ell + 1)}C_{k-1}C_{\ell+1} }$$

and

$$\displaystyle\begin{array}{rcl} \frac{(2k - 1)(\ell+2)} {(k + 1)(2\ell + 1)}& <& 1 {}\\ \Leftrightarrow \ \ \ \ 3k - 3& <& 3\ell {}\\ \Leftrightarrow \ \ \ \ k& \leq & \ell\text{ since }k\text{ and }\ell\text{ are integers.} {}\\ \end{array}$$

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Fact A.5

For $m \in \mathbb{N}$ ,

$$\displaystyle{ (2m - 1)\left (2C_{m-1} -\frac{1} {2}C_{m}\right ) = \frac{3} {2}C_{m} }$$

Proof

Employ Fact A.2 for expressing C _m−1 in terms of C _m .

$$\displaystyle\begin{array}{rcl} (2m - 1)\left (2C_{m-1} -\frac{1} {2}C_{m}\right )& =& (2m - 1)\left (2\left (\frac{1} {2} \frac{m + 1} {2m - 1}C_{m}\right ) -\frac{1} {2}C_{m}\right ) {}\\ & =& (2m - 1)\left (\frac{2(m + 1) - (2m - 1)} {2(2m - 1)} \right )C_{m} {}\\ & =& \frac{3} {2}C_{m} {}\\ \end{array}$$

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Fact A.6

For $k,\ell\in \mathbb{N}$ ,

$$\displaystyle{ \frac{2k - 1} {2k} \frac{C_{k-1}} {C_{k}} < \frac{C_{\ell-1}} {C_{\ell}} \mathit{\mbox{ iff }}2\ell - 1 < 3k }$$

Proof

$$\displaystyle\begin{array}{rcl} \frac{2k - 1} {2k} \frac{C_{k-1}} {C_{k}} & <& \frac{C_{\ell-1}} {C_{\ell}} {}\\ \Leftrightarrow \ \ \ \ \frac{2k - 1} {2k} \frac{k + 1} {2(2k - 1)}& <& \frac{\ell+1} {2(2\ell - 1)}\mbox{ due to Fact A.2} {}\\ \Leftrightarrow \ \ \ \ 2\ell - 1& <& 3k {}\\ \end{array}$$

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Fact A.7

For all $m \in \mathbb{N}$ , m ≥ 2,

$$\displaystyle{ C_{m} + C_{m-1} - 2C_{m-2} = \frac{9} {8}C_{m}\left (1 + \Theta (1/m^{2})\right )\ . }$$

Proof

$$\displaystyle\begin{array}{rcl} C_{m} + C_{m-1} - 2C_{m-2}& =& C_{m} + \frac{1} {2} \frac{m + 1} {2m - 1}C_{m} - 2(\frac{1} {2} \frac{m + 1} {2m - 1} \cdot \frac{1} {2} \frac{m} {2m - 3})C_{m} {}\\ & =& C_{m}\,\left (\frac{9m^{2} - 18m + 3} {8m^{2} - 16m + 6}\right ) {}\\ & =& \frac{9} {8}C_{m}\left (1 - \frac{5} {3(4m^{2} - 8m + 3)}\right ) {}\\ & =& \frac{9} {8}C_{m}\,(1 + \Theta (1/m^{2})) {}\\ \end{array}$$

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