the other block. If no two blocks cross, then the partition is said to be [*noncrossing*]{}. A [*matching*]{} is a partition in which every block has cardinality 2.

A noncrossing partition can be represented by placing the numbers $a,a+1,\dots,b$ around a circle and connecting numbers in the same block, as in Figure \[NC-example-figure\].

If the curve $C_\theta(f)$ is nonsingular, then the matching ${M}(f,\theta)$ is noncrossing by the Jordan Curve Theorem.

The theory of noncrossing partitions comprises a substantial chapter in modern combinatorics; see [@Simion] for a comprehensive survey. In the present study, we will be concerned most with noncrossing matchings on the sets $${{\bf E}_n}=\{0,2,4,\dots,4n-2\}\qquad\text{and}\qquad {{\bf O}_n}= \{1,3,\dots,4n-1\},$$ where $n$ is a positive integer.

It is well-known (and elementary to verify) that the number of matchings on ${{\bf E}_n}$ (resp. ${{\bf O}_n}$) is $(2n-1)(2n-3)\cdots(3)(1) = \frac{(2n)!}{2^nn!}$, and that the number of noncrossing matchings is the Catalan number $\frac{1}{n+1}\binom{2n}{n}$.

\[lemma:parity-NCM\] Let $M$ be a matching on $[0,2n-1]$. If $M$ is noncrossing, then exactly one member of each pair $\{i,j\}$ is even.

Suppose that $i < j$ and $i\equiv j \pmod{2}$. Then the sets $$X = [i+1,j-1], \qquad Y = [0,i-1] \cup [j+1,2n-1]$$ both have odd cardinality, so some $x\in X$ is 