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Zap. Nauchn. Sem. POMI, 2005, Volume 325, Pages 83–102 (Mi znsl351)  

This article is cited in 35 scientific papers (total in 35 papers)

Exchangeable Gibbs partitions and Stirling triangles

A. V. Gnedina, J. Pitmanb

a Utrecht University
b University of California, Berkeley

Abstract: For two collections of nonnegative and suitably normalised weights $W=(W_j)$ and $V=(V_{n,k})$, a probability distribution on the set of partitions of the set $\{1,\ldots,n\}$ is defined by assigning to a generic partition $\{A_j, j\leq k\}$ the probability $V_{n,k} W_{|A_1|}\cdots W_{|A_k|}$, where $|A_j|$ is the number of elements of $A_j$. We impose constraints on the weights by assuming that the resulting random partitions $\Pi_n$ 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 $\alpha\in[-\infty,1]$. The case $\alpha=1$ is trivial, and for each value of $\alpha\ne 1$ 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 generalised Stirling triangle. In particular, we show that the boundary is discrete for $-\infty\le\alpha<0$ and continuous for $0\le\alpha<1$. For $\alpha\le 0$ the extremes correspond to the members of the Ewens–Pitman family of random partitions indexed by $(\alpha,\theta)$, while for $0<\alpha<1$ the extremes are obtained by conditioning an $(\alpha,\theta)$-partition on the asymptotics of the number of blocks of $\Pi_n$ as $n$ tends to infinity.

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English version:
Journal of Mathematical Sciences (New York), 2006, 138:3, 5674–5685

Bibliographic databases:

UDC: 519.217.72, 519.217.74
Received: 25.04.2005

Citation: A. V. Gnedin, J. Pitman, “Exchangeable Gibbs partitions and Stirling triangles”, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XII, Zap. Nauchn. Sem. POMI, 325, POMI, St. Petersburg, 2005, 83–102; J. Math. Sci. (N. Y.), 138:3 (2006), 5674–5685

Citation in format AMSBIB
\by A.~V.~Gnedin, J.~Pitman
\paper Exchangeable Gibbs partitions and Stirling triangles
\inbook Representation theory, dynamical systems, combinatorial and algoritmic methods. Part~XII
\serial Zap. Nauchn. Sem. POMI
\yr 2005
\vol 325
\pages 83--102
\publ POMI
\publaddr St.~Petersburg
\jour J. Math. Sci. (N. Y.)
\yr 2006
\vol 138
\issue 3
\pages 5674--5685

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