In general, to find maps transporting statistics in a well-behaved way, it is useful to try [FindStat][1].  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:

1. a statistic $stat$ on (possibly different) combinatorial objects, and
2. 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,...

  [1]: http://www.findstat.org