  Parametric families in the centre ${\bf Z}({\bf C}[S_n])$ of the group
algebra of the symmetric group are obtained by identifying the indeterminates
in the generating function for Macdonald polynomials as commuting Jucys-Murphy
elements. Their eigenvalues provide coefficients in the double Schur function
expansion of 2D Toda $\tau$-functions of hypergeometric type. Expressing these
in the basis of products of power sum symmetric functions, the coefficients may
be interpreted geometrically as parametric families of quantum Hurwitz numbers,
enumerating weighted branched coverings of the Riemann sphere. Combinatorially,
they give quantum weighted sums over paths in the Cayley graph of $S_n$
generated by transpositions. Dual pairs of bases for the algebra of symmetric
functions with respect to the scalar product in which the Macdonald polynomials
are orthogonal provide both the geometrical and combinatorial significance of
these quantum weighted enumerative invariants.
