  We generalize a construction of Dunkl, obtaining a wide class intertwining
functions on the symmetric group and a related family of multidimensional Hahn
polynomials. Following a suggestion of Vilenkin and Klymik, we develop a
tree-method approach for those intertwining functions. We also give a group
theoretic proof of the relation between Hahn polynomials and Clebesh-Gordan
coefficients, given analytically by Koornwinder and by Nikiforov,
Smorodinski\u{i} and Suslov. Such relation is also extended to the
multidimensional case.
