In general, to find maps transporting statistics in a well-behaved way, it is useful to try FindStat. In the case at hand, go to

http://www.findstat.org/StatisticsDatabase/St000021/

(which is the statistic "number of descents of a permutation") and click on "Search for values". After a short while, you will be presented with a list of candidates, each of the following type:

- a statistic $stat$ on (possibly different) combinatorial objects, and
- a map $\phi$ such that
$$
des(\pi) = stat(\phi(\pi))
$$
(possibly $\phi$ is in fact a composition of several maps)

You then only have to check which of candidates have maps that are bijective. Furthermore, you will have to check that not only the *number of descents* but also the descent set itself is preserved.

In the case at hand, Ira's example of standard Young tableaux is found, there is possibly a well behaved bijection to increasing trees, to ordered trees,...

As Christian points out in the comment below, it is also possible to provide values only for a subset of permutations, in your case for fixed point free involutions.

Yet another possibility is to use a collection of objects built into FindStat that fits your problem better, namely http://www.findstat.org/StatisticFinder/PerfectMatchings.

The drawback of the latter two approaches is that you have to enter the values manually (or generate them with a computer program as below and use the "free" box).

```
for n in range(1,4):
for pi in PerfectMatchings(2*n):
print pi, "=>", pi.to_permutation().number_of_descents()
```