The number of cycles in the breakpoint graph of a permutation.
The breakpoint graph of a permutation $\pi_1,\dots,\pi_n$ is the directed, bicoloured graph with vertices $0,\dots,n$, a grey edge from $i$ to $i+1$ and a black edge from $\pi_i$ to $\pi_{i-1}$ for $0\leq i\leq n$, all indices taken modulo $n+1$.
This graph decomposes into alternating cycles, which this statistic counts.
The distribution of this statistic on permutations of $n-1$ is, according to [cor.1, 5] and [eq.6, 6], given by
$$\frac{1}{n(n+1)}((q+n)_{n+1}-(q)_{n+1}),$$
where $(x)_n=x(x-1)\dots(x-n+1)$.

Sends a Dyck path of semilength n to the noncrossing perfect matching given by matching an up-step with the corresponding down-step.
This is, for a Dyck path $D$ of semilength $n$, the perfect matching of $\{1,\dots,2n\}$ with $i < j$ being matched if $D_i$ is an up-step and $D_j$ is the down-step connected to $D_i$ by a tunnel.

This map sends a permutation $\pi$ to $\pi^{-1} \star \pi$ where $\star$ denotes the Demazure product on permutations.
This map is a surjection onto the set of involutions, i.e., the set of permutations $\pi$ for which $\pi = \pi^{-1}$.

The fixed-point-free permutation with deficiencies given by the perfect matching, no alignments and no inversions between exceedences.
Put differently, the exceedences form the unique non-nesting perfect matching whose openers coincide with those of the given perfect matching.