In this project, we'll focus on questions related to permutation statistics. These are nonnegative-integer-valued functions on permutations; classical examples of statistics on a permutation p include its descent number des(p) (the number of i such that p(i)>p(i+1)), the number of inversions inv(p) (the number of i < j such that p(i)>p(j)), and the major index maj(p) (the sum of the values i such that p(i)>p(i+1)). One important problem is the determination of the distribution of a statistic (or tuples of statistics)---i.e., the number of permutations for which the statistic takes a certain value---over all permutations of a given length. A number of interesting results have been obtained showing that certain statistics (or tuples of statistics) have the same distribution; see  for one recent example of this that also provides a way to decompose classical statistics into sums of other statistics.
If you are interested in this, feel free to e-mail me with questions or comments.