A polytope related to empirical distributions, plane trees, parking functions, and the associahedron

June, 1999
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Jim Pitman and Richard Stanley
Discrete and Computational Geometry 27: 603-634 (2002)

The volume of the n-dimensional polytope of all (y_1, ... , y_n) with y_i > 0 and y_1 + ... + y_i < x_1 + ... + x_i for all i for arbitrary (x_1, ... , x_n) with x_i > 0 for all i defines a polynomial in variables x_i 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 different polytopal subdivisions. The first 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 configuration of these chambers provides a representation of another polytope with many applications, the associahedron.

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