Several variants of the Dumont differential system and permutation statistics

The Dumont differential system on the Jacobi elliptic functions was introduced by Dumont (1979) and was extensively studied by Dumont, Viennot, Flajolet and so on. In this paper, we first present a labeling scheme for the cycle structure of permutations. We then introduce two types of Jacobi-pairs of differential equations. We present a general method to derive the solutions of these differential equations. As applications, we present some characterizations for several permutation statistics.