Concrete Math
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We define so-called poset-polynomials of a graded poset and use it to give an explicit and general description of the combinatorial numbers in Schlafli's (combinatorial) reduction formula. For simplicial and simple polytopes these combinatorial numbers can be written as functions of the numbers of faces of the polytope and the tangent numbers. We use the constructed formulas to determine the volume of 4-dimensional Coxeter polytopes.