are interested primarily in the case when no extra crossings occur.

A [*basketball of order $n$*]{} (or simply an [*$n$-basketball*]{}) is a bimatching $B=({{B^{\bf e}}},{{B^{\bf o}}})$ of order $n$ in which the matchings ${{B^{\bf e}}},{{B^{\bf o}}}$ are both noncrossing, and each pair $e\in{{B^{\bf e}}}$ crosses exactly one pair $o\in{{B^{\bf o}}}$. The pair of pairs $e,o$ is called a [*quartet*]{}. The set of all $n$-basketballs is denoted ${{\mathcal{B}}}_n$.

The bimatching ${B}(f,\alpha,\beta)$ of Example \[ex:bimatch\] is a basketball of order $n$. Indeed, every crossing between an even pair (i.e., an arc of $C_\alpha(f)$) and an odd pair (i.e., an arc of $C_\beta(f)$) corresponds to a root of $f$, and there are exactly $n$ roots.

\[thm:count-bballs\] The number of basketballs of order $n$ is $\frac{1}{3n+1}\binom{4n}{n}$.

By a special case of [@Edelman Lemma 4.1], the given quantity enumerates the set ${{\mathcal{Q}}}_n$ of noncrossing partitions of $[0,4n-1]$ into $n$ blocks of size 4. We will exhibit a bijection $\phi:{{\mathcal{B}}}_n\to{{\mathcal{Q}}}_n$.

Let $B \in {{\mathcal{B}}}_n$. Define a partition $\phi(B)$ of $[0,4n-1]$ into $n$ blocks of size 4 by replacing each quartet $\{i,j\},\{k,\ell\}$ with the block $\{i,j,k,\ell\}$. It is elementary to deduce from the definition of a basketball that $\phi(B)$ is noncrossing, hence belongs to ${{\mathcal{Q}}}_n$. It is 