  The U-polynomial, the polychromate and the symmetric function generalization
of the Tutte polynomial due to Stanley are known to be equivalent in the sense
that the coefficients of any one of them can be obtained as a function of the
coefficients of any other. The definition of each of these functions suggests a
natural way in which to generalize them which also captures Tutte's universal
V-functions as a specialization. We show that the equivalence remains true for
the extended functions thus answering a question raised by Dominic Welsh.
