A Polytope Related to Empirical Distributions, Plane Trees, Parking Functions, and the Associahedron

The volume of the n-dimensional polytope n(x) := fy 2 R n : yi 0 and y1 + + yi x1 + + xi for all 1 i ng for arbitrary x := (x1; : : : ; xn) with xi > 0 for all i de nes a polynomial in variables xi which admits a number of interpretations, in terms of empirical distributions, plane partitions, and parking functions. We interpret the terms of this polynomial as the volumes of chambers in two di erent polytopal subdivisions of n(x). The rst of these subdivisions generalizes to a class of polytopes called sections of order cones. In the second subdivision, the chambers are indexed in a natural way by rooted binary trees with n + 1 vertices, and the con guration of these chambers provides a representation of another polytope with many applications, the associahedron. Research supported in part by NSF grant 97-03961 Research supported in part by NSF grant 95-00714