A Nekrasov–Okounkov type formula for C˜

In 2008, Han rediscovered an expansion of powers of Dedekind η   function attributed to Nekrasov and Okounkov (which was actually first proved the same year by Westbury) by using a famous identity of Macdonald in the framework of type A˜ affine root systems. In this paper, we obtain new combinatorial expansions of powers of η  , in terms of partition hook lengths, by using the Macdonald identity in type C˜ and a new bijection between vectors with integral coordinates and a subset of t  -cores for integer partitions. As applications, we derive a symplectic hook formula and an unexpected relation between the Macdonald identities in types C˜, B˜, and BC˜. We also generalize these expansions through the Littlewood decomposition and deduce in particular many new weighted generating functions for subsets of integer partitions and refinements of hook formulas.