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 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 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 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 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.

scholarly journals On the least common multiple of random q-integers

2021 ◽  
Vol 7 (1) ◽  
Author(s):  
Carlo Sanna
Keyword(s):  
Positive Integer ◽  
Asymptotic Formula ◽  
Probabilistic Model ◽  
Expected Value ◽  
Random Set ◽  
Common Multiple ◽  
Least Common Multiple

AbstractFor every positive integer n and for every $$\alpha \in [0, 1]$$ α ∈ [ 0 , 1 ] , let $${\mathcal {B}}(n, \alpha )$$ B ( n , α ) denote the probabilistic model in which a random set $${\mathcal {A}} \subseteq \{1, \ldots , n\}$$ A ⊆ { 1 , … , n } is constructed by picking independently each element of $$\{1, \ldots , n\}$$ { 1 , … , n } with probability $$\alpha $$ α . Cilleruelo, Rué, Šarka, and Zumalacárregui proved an almost sure asymptotic formula for the logarithm of the least common multiple of the elements of $${\mathcal {A}}$$ A .Let q be an indeterminate and let $$[k]_q := 1 + q + q^2 + \cdots + q^{k-1} \in {\mathbb {Z}}[q]$$ [ k ] q : = 1 + q + q 2 + ⋯ + q k - 1 ∈ Z [ q ] be the q-analog of the positive integer k. We determine the expected value and the variance of $$X := \deg {\text {lcm}}\!\big ([{\mathcal {A}}]_q\big )$$ X : = deg lcm ( [ A ] q ) , where $$[{\mathcal {A}}]_q := \big \{[k]_q : k \in {\mathcal {A}}\big \}$$ [ A ] q : = { [ k ] q : k ∈ A } . Then we prove an almost sure asymptotic formula for X, which is a q-analog of the result of Cilleruelo et al.

Random set limit theorems

1979 ◽  
Vol 11 (02) ◽  
pp. 281-282 ◽  
Author(s):  
Noel Cressie
Keyword(s):  
Limit Theorems ◽  
Random Set
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