  We present a simplified variant of Biane's bijection between permutations and
3-colored Motzkin paths with weight that keeps track of the inversion number,
excedance number and a statistic so-called depth of a permutation. This
generalizes a result by Guay-Paquet and Petersen about a continued fraction of
the generating function for depth on the permutations of n elements. In terms
of weighted Motzkin path, we establish an involution on the permutations that
reverses the parities of depth and excedance numbers simultaneously, which
proves that the numbers of permutations with even and odd depth (excedance
numbers, respectively) are equal if n is even and differ by the tangent number
if n is odd. Moreover, we present some interesting sign-imbalance results on
permutations and derangements, refined with respect to depth and excedance
numbers.
