|Date||Apr 02, 2015|
To explain Ramanujan's integer partition function congruences, Dyson's rank and Andrews-Garvan's crank have been introduced. The generating functions for these two partition statistics are typical examples of mock Jacobi forms and Jacobi forms, and have been scorches of numerous researches in number theory and combinatorics. In particular, studying how their distributions differ is one of main themes in the theory of partitions. In this talk, we introduce recent results on their distributions with emphasizing on roles of q-series, combinatorial methods, and modular forms.