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arXiv · 2309.11292

Polychromatic Partitions, Kingman Theory, and Ewens Sampling Formula

Lorenzo Dello Schiavo·Filippo Quattrocchi
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Abstract

A polychromatic partition is a partition of a set of colored (marked) elements in which each block is recorded only by the number of elements of each color in that block. We introduce a notion of polychromatic partition structure, i.e. a family of random polychromatic partitions consistent under deletion of elements uniformly at random, simultaneously extending (standard) partition structures, multipartition structures, and partition structures of partially exchangeable type. As an example thereof, we define a polychromatic analogue of the celebrated Ewens Sampling Formula, proving polychromatic versions of Kingman's characterization and of Hoppe's urn model. As our main result, we give a complete characterization of polychromatic partition structures, proving a triple affine homeomorphism of Bauer simplices among: polychromatic partition structures, harmonic densities for the polychromatic Hoppe urn, and probability measures on a polychromatic Kingman simplex. Additionally, we prove that this new simplex is homeomorphic to the Martin boundary of the polychromatic Hoppe urn inside its Martin compactification. We discuss various applications including closed non-recursive non-iterative expressions for multivariate moments of several random measures in terms of cycle index polynomials whose monomials are indexed by polychromatic partitions.

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Lorenzo Dello Schiavo, Filippo Quattrocchi. 2026-08-30. Polychromatic Partitions, Kingman Theory, and Ewens Sampling Formula. https://arxiv.org/abs/2309.11292

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