Applications in Computer Sciences of Partition Functions A partition function computes the number of ways an integer $n$ can be represented as the sum of $m$ other integers. For some value $n$, we have a partition function $p(n)$.

These were studied extensively in mathematics by MacMahon, Hardy, Littlewood and Ramanujan. Rademacher provided a way to approximating $p(n)$, improving on a formula by hardy and Littlewood. More recently, K. Ono and A. Wiles have come up with new interesting results. 

Are there applications of partition functions in Computer Sciences, perhaps in the theory of elliptic curve cryptography or complexity?
there is a connection between partition functions and counting possible edge and vertex colorings in graphs, but its very abstract. see eg Graph parameters and invariants of the orthogonal group, Regts
there are applications of partition functions to simulating Ising models for Spin glasses in physics. spin glasses have been shown to have deep connections to NP complete phase transitions.