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Generalized Z_k^(r/2) parafermionic theories - characterized by the dimension(r/2)(1-1/k) of the basic parafermionic field - provide potentially interestingquantum-Hall trial wavefunctions. Such wavefunctions reveal a W_k structure.This suggests the equivalence of (a subclass of) the Z_k^(r/2) models and theW_k(k+1,k+r) ones. This is demonstrated here for r=3 (the gaffnian series). Theagreement of the parafermionic and the W spectra relies on the priordetermination of the field identifications in the parafermionic case.

The W_k structure of the Z_k^(3/2) models