Parking functions, tree depth and factorizations of the full cycle into transpositions

Consider the set of minimal factorizations of the canonical full cycle in the symmetric group on n + 1 symbols. In 2002, Biane found a remarkably simple bijection from this set to the set of parking functions of length n; the bijection maps a factorization to the sequence consisting of the smallest element from each transposition. Thus, it is utterly trivial to find the image of a factorization in this map, but reversing this map requires much more work. Furthermore, as far as parking functions are concerned, it appears that the largest element in each transposition can be discarded. We show, however, that the sequence given by the largest element of each transposition also displays some interesting properties. In particular, the natural area statistics on this sequence and the parking function together correspond to two natural statistics on trees: the inversion number (this is well known) and the non-inversion number. This allows us to present a bivariate generating series, which is a natural generalization of the univariate inversion series. We also give a number of related results