Bijective enumeration and sign-imbalance for permutation depth and excedances

We present a simplified exposition of a variant of Foata–Zeilberger bijection between permutations and 3-colored Motzkin paths with weight that keeps track of multiple statistics of a permutation. Involving a statistic so-called depth, we present some bijective generalizations of a result by Guay-Paquet and Petersen about a continued fraction of the generating function for depth on the symmetric group ℭ_n of permutations. In terms of weighted Motzkin path, we establish an involution on ℭ_n 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. We also show that the nesting numbers and crossing numbers are symmetrically distributed over the permutations with prescribed depth and excedance numbers. Moreover, we present some interesting sign-imbalance results on permutations and derangements, refined with respect to depth and excedance numbers.