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Journal of Mathematical Sciences

, Volume 138, Issue 3, pp 5674–5685 | Cite as

Exchangeable Gibbs partitions and Stirling triangles

  • A. Gnedin
  • J. Pitman
Article

Abstract

For two collections of nonnegative and suitably normalized weights W = (Wj) and V = (Vn,k), a probability distribution on the set of partitions of the set {1, …, n} is defined by assigning to a generic partition {Aj, j ≤ k} the probability Vn,k \(V_{n,k} W_{\left| {A_1 } \right|} \cdots W_{\left| {A_k } \right|} \), where |Aj| is the number of elements of Aj. We impose constraints on the weights by assuming that the resulting random partitions Π n of [n] are consistent as n varies, meaning that they define an exchangeable partition of the set of all natural numbers. This implies that the weights W must be of a very special form depending on a single parameter α ∈ [− ∞, 1]. The case α = 1 is trivial, and for each value of α ≠ = 1 the set of possible V-weights is an infinite-dimensional simplex. We identify the extreme points of the simplex by solving the boundary problem for a generalized Stirling triangle. In particular, we show that the boundary is discrete for − ∞ ≤ α < 0 and continuous for 0 ≤ α < 1. For α ≤ 0 the extremes correspond to the members of the Ewens-Pitman family of random partitions indexed by (α,θ), while for 0 < α < 1 the extremes are obtained by conditioning an (α,θ)-partition on the asymptotics of the number of blocks of Πn as n tends to infinity. Bibliography: 29 titles.

Keywords

Random Partition Stirling Number Extreme Element Regular Path Gibbs Form 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • A. Gnedin
    • 1
  • J. Pitman
    • 2
  1. 1.Utrecht UniversityThe Netherlands
  2. 2.University of California at BerkeleyUSA

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