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We explore some connections between vectors of integers and integerpartitions seen as bi-infinite words. This methodology enables us on the onehand to obtain enumerations connecting products of hook lengths and vectors ofintegers. This yields on the other hand a combinatorial interpretation of theMacdonald identities for affine root systems of the $7$ infinite families interms of Schur functions, symplectic and special orthogonal Schur functions.From these results, we are able to derive $q$-Nekrasov--Okounkov formulasassociated to each type. The latter for limit cases of $q$ yieldNekrasov--Okounkov type formulas corresponding to all the specializations givenby Macdonald.

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