scholarly journals Reversible Markov structures on divisible set partitions

2015 ◽  
Vol 52 (3) ◽  
pp. 622-635
Author(s):  
Harry Crane ◽  
Peter McCullagh
Keyword(s):  
Markov Chains ◽  
Random Process ◽  
Random Set ◽  
Set Partitions ◽  
Chinese Restaurant ◽  
Partition Structure ◽  
Block Sizes

We study k-divisible partition structures, which are families of random set partitions whose block sizes are divisible by an integer k = 1, 2, …. In this setting, exchangeability corresponds to the usual invariance under relabeling by arbitrary permutations; however, for k > 1, the ordinary deletion maps on partitions no longer preserve divisibility, and so a random deletion procedure is needed to obtain a partition structure. We describe explicit Chinese restaurant-type seating rules for generating families of exchangeable k-divisible partitions that are consistent under random deletion. We further introduce the notion of Markovian partition structures, which are ensembles of exchangeable Markov chains on k-divisible partitions that are consistent under a random process of Markovian deletion. The Markov chains we study are reversible and refine the class of Markov chains introduced in Crane (2011).

scholarly journals Reversible Markov structures on divisible set partitions

2015 ◽  
Vol 52 (03) ◽  
pp. 622-635
Author(s):  
Harry Crane ◽  
Peter McCullagh
Keyword(s):  
Markov Chains ◽  
Random Process ◽  
Random Set ◽  
Set Partitions ◽  
Chinese Restaurant ◽  
Partition Structure ◽  
Block Sizes

We studyk-divisible partition structures, which are families of random set partitions whose block sizes are divisible by an integerk= 1, 2, …. In this setting, exchangeability corresponds to the usual invariance under relabeling by arbitrary permutations; however, fork> 1, the ordinary deletion maps on partitions no longer preserve divisibility, and so a random deletion procedure is needed to obtain a partition structure. We describe explicit Chinese restaurant-type seating rules for generating families of exchangeablek-divisible partitions that are consistent under random deletion. We further introduce the notion ofMarkovian partition structures, which are ensembles of exchangeable Markov chains onk-divisible partitions that are consistent under a random process ofMarkovian deletion. The Markov chains we study are reversible and refine the class of Markov chains introduced in Crane (2011).

scholarly journals On the maximal multiplicity of block sizes in a random set partition

2020 ◽  
Vol 56 (3) ◽  
pp. 867-891
Author(s):  
Ljuben R. Mutafchiev ◽  
Mladen Savov
Keyword(s):  
Random Set ◽  
Set Partition ◽  
Maximal Multiplicity ◽  
Block Sizes

scholarly journals Regenerative Partition Structures

10.37236/1869 ◽  
2005 ◽  
Vol 11 (2) ◽  
Author(s):  
Alexander Gnedin ◽  
Jim Pitman
Keyword(s):  
Random Number ◽  
Probability Distributions ◽  
Random Allocation ◽  
Random Set ◽  
Random Partition ◽  
Independent Increments ◽  
Increasing Process ◽  
Composition Structure ◽  
Partition Structure ◽  
Two Parameters

A partition structure is a sequence of probability distributions for $\pi_n$, a random partition of $n$, such that if $\pi_n$ is regarded as a random allocation of $n$ unlabeled balls into some random number of unlabeled boxes, and given $\pi_n$ some $x$ of the $n$ balls are removed by uniform random deletion without replacement, the remaining random partition of $n-x$ is distributed like $\pi_{n-x}$, for all $1 \le x \le n$. We call a partition structure regenerative if for each $n$ it is possible to delete a single box of balls from $\pi_n$ in such a way that for each $1 \le x \le n$, given the deleted box contains $x$ balls, the remaining partition of $n-x$ balls is distributed like $\pi_{n-x}$. Examples are provided by the Ewens partition structures, which Kingman characterised by regeneration with respect to deletion of the box containing a uniformly selected random ball. We associate each regenerative partition structure with a corresponding regenerative composition structure, which (as we showed in a previous paper) is associated in turn with a regenerative random subset of the positive halfline. Such a regenerative random set is the closure of the range of a subordinator (that is an increasing process with stationary independent increments). The probability distribution of a general regenerative partition structure is thus represented in terms of the Laplace exponent of an associated subordinator, for which exponent an integral representation is provided by the Lévy-Khintchine formula. The extended Ewens family of partition structures, previously studied by Pitman and Yor, with two parameters $(\alpha,\theta)$, is characterised for $0 \le \alpha < 1$ and $\theta >0$ by regeneration with respect to deletion of each distinct part of size $x$ with probability proportional to $(n-x)\tau+x(1-\tau)$, where $\tau = \alpha/(\alpha+\theta)$.

Random Set Partitions

1994 ◽  
Vol 7 (3) ◽  
pp. 419-436 ◽  
Author(s):  
William M. Y. Goh ◽  
Eric Schmutz
Keyword(s):  
Random Set ◽  
Set Partitions

scholarly journals Average Values of some Z-Parameters in a Random Set Partition

10.37236/715 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Anisse Kasraoui
Keyword(s):  
Asymptotic Formulas ◽  
Random Set ◽  
Set Partition ◽  
Set Partitions

We find exact and asymptotic formulas for the average values of several statistics on set partitions: of Carlitz's $q$-Stirling distributions, of the numbers of crossings in linear and circular representations of set partitions, of the numbers of overlappings and embracings, and of the numbers of occurrences of a 2-pattern.

scholarly journals Relational exchangeability

2019 ◽  
Vol 56 (01) ◽  
pp. 192-208 ◽  
Author(s):  
Harry Crane ◽  
Walter Dempsey
Keyword(s):  
Network Analysis ◽  
General Theory ◽  
General Class ◽  
Structure Theorem ◽  
Combinatorial Structure ◽  
Interaction Data ◽  
Random Set ◽  
Set Partitions ◽  
Alternative Perspective ◽  
Random Combinatorial Structure

AbstractA relationally exchangeable structure is a random combinatorial structure whose law is invariant with respect to relabeling its relations, as opposed to its elements. Historically, exchangeable random set partitions have been the best known examples of relationally exchangeable structures, but the concept now arises more broadly when modeling interaction data in modern network analysis. Aside from exchangeable random partitions, instances of relational exchangeability include edge exchangeable random graphs and hypergraphs, path exchangeable processes, and a range of other network-like structures. We motivate the general theory of relational exchangeability, with special emphasis on the alternative perspective it provides and its benefits in certain applied probability problems. We then prove a de Finetti-type structure theorem for the general class of relationally exchangeable structures.

scholarly journals The Asymptotic Number of Set Partitions with Unequal Block Sizes

10.37236/1434 ◽  
1998 ◽  
Vol 6 (1) ◽  
Author(s):  
A. Knopfmacher ◽  
A. M. Odlyzko ◽  
B. Pittel ◽  
L. B. Richmond ◽  
D. Stark ◽  
...  
Keyword(s):  
Asymptotic Behavior ◽  
Generating Function ◽  
Direct Approach ◽  
Set Partition ◽  
Random Partition ◽  
Set Partitions ◽  
Analytic Methods ◽  
Asymptotic Number ◽  
Block Sizes ◽  
Element Set

The asymptotic behavior of the number of set partitions of an $n$-element set into blocks of distinct sizes is determined. This behavior is more complicated than is typical for set partition problems. Although there is a simple generating function, the usual analytic methods for estimating coefficients fail in the direct approach, and elementary approaches combined with some analytic methods are used to obtain most of the results. Simultaneously, we obtain results on the shape of a random partition of an $n$-element set into blocks of distinct sizes.

Markov models—Markov chains

2019 ◽  
Vol 16 (8) ◽  
pp. 663-664 ◽  
Author(s):  
Jasleen K. Grewal ◽  
Martin Krzywinski ◽  
Naomi Altman
Keyword(s):  
Markov Chains ◽  
Markov Models
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