Several variants of the Dumont differential system and permutation   statistics
  The Dumont differential system on the Jacobi elliptic functions was introduced by Dumont (Math Comp, 1979, 33: 1293--1297) 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. 
Shi-Mei Ma, Toufik Mansour, David G.L. Wang, Yeong-Nan Yeh
math.CO