Is there a bijection of permutations onto mathematical objects that preserve information about descents? $\omega \in S_n$ is an FPFI (fixed point free involution) (also called a matching) if $\omega^2=1$ and $\omega(i) \neq i$ for all $i$.
For $\omega \in S_n$, a descent occurs at $i$ if $\omega(i+1) < \omega(i)$. For example, $(1 \, 3)(2 \, 4) \in S_4$, when written as $3412$, has one descent at $i=2$.
I'm curious as to what kind of mathematical objects there are that FPFIs be bijected onto that preserves the descent set, if any? Or in general, if there are objects that permutations can be bijected onto that preserve the information about descents.
 A: 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()

