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.
Bibliographical noteFunding Information:
This work was supported by National Natural Science Foundation of China (Grant No. 11401083), Natural Science Foundation of Hebei Province (Grant No. A2017501007), the Fundamental Research Funds for the Central Universities (Grant No. N152304006) and Taiwan ?National? Science Council (Grant No. 104-2115-M-001-010). This work was finished while Yeh was visiting the School of Mathematical Sciences, Dalian University of Technology, Dalian, China.
© 2018, Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature.
- Dumont differential system
- Jacobi elliptic functions
- context-free grammars
- permutation statistics
ASJC Scopus subject areas
- Mathematics (all)