We introduce a new class of combinatorially defined rational functions and apply them to deduce explicit formulae for local ideal zeta functions associated to the members of a large class of nilpotent Lie rings which contains the free class-2-nilpotent Lie rings and is stable under direct products. Our results unify and generalize a substantial number of previous computations. We show that the new rational functions, and thus also the local zeta functions under consideration, enjoy a self-reciprocity property, expressed in terms of a functional equation upon inversion of variables. We establish a conjecture of Grunewald, Segal, and Smith on the uniformity of normal zeta functions of finitely generated free class-2-nilpotent groups.
 We produce explicit formulae for various ideal zeta functions associated to the members of an infinite family of class- @math -nilpotent Lie rings, introduced in [8], in terms of Igusa functions. As corollaries we obtain information about analytic properties of global ideal zeta functions, local functional equations, topological, reduced, and graded ideal zeta functions, as well as representation zeta functions for the unipotent group schemes associated to the Lie rings in question. The pro-isomorphic zeta function ( ^ _ (s) ) of a finitely generated nilpotent group ( ) is a Dirichlet generating function that enumerates finite-index subgroups whose profinite completion is isomorphic to that of ( ). Such zeta functions can be expressed as Euler products of p-adic integrals over the ( Q _p )-points of an algebraic automorphism group associated to ( ). In this way they are closely related to classical zeta functions of algebraic groups over local fields. A ski is provided on its under surface with a longitudinal central part having a smoother surface configuration than the adjacent surfaces and this ski is to be used on a substantially hard smooth surface especially on an inclined belt which can be adjusted as to speed and inclination.