  We explore some connections between vectors of integers and integer
partitions seen as bi-infinite words. This methodology enables us to give a
combinatorial interpretation of the Macdonald identities for affine root
systems of the seven infinite families in terms of symplectic and special
orthogonal Schur functions. From these results, we are able to derive
$q$-Nekrasov--Okounkov formulas associated to each family. Nevertheless we only
give results for types $\tilde{C}$ and $\tilde{C}^{\vee}$, and give a sketch of
the proof for type $\tilde{C}$.
