Some combinatorial interpretations of the Macdonald identities for affine root systems and Nekrasov--Okounkov type formulas

We explore some connections between vectors of integers and integer partitions seen as bi-infinite words. This methodology enables us on the one hand to obtain enumerations connecting products of hook lengths and vectors of integers. This yields on the other hand a combinatorial interpretation of the Macdonald identities for affine root systems of the $7$ infinite families in terms of Schur functions, symplectic and special orthogonal Schur functions. From these results, we are able to derive $q$-Nekrasov–Okounkov formulas associated to each type. The latter for limit cases of $q$ yield Nekrasov–Okounkov type formulas corresponding to all the specializations given by Macdonald.