Random Combinatorial Structures

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Probability and Statistics".

Deadline for manuscript submissions: closed (30 September 2023) | Viewed by 7859

Special Issue Editors


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Guest Editor
Department of Statistics, School of Mathematics, University of Leeds, Leeds LS2 9JT, UK
Interests: theoretical and applied probability; random processes; statistics; modelling of extreme values; statistical physics; mathematical physics; probabilistic combinatorics

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Guest Editor
Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, UK
Interests: probability; random matrix theory; number theory; representation theory; function theory; combinatorics

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Guest Editor
School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China
Interests: probability limit theory with applications; random integer partitions; random matrix theory; percolation theory; random growth models

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Guest Editor
Department of Probability Theory and Mathematical Statistics, Mathematics and Mechanics Faculty, St. Petersburg State University, 198504 St. Petersburg, Russia
Interests: probability theory; stochastic processes; probabilistic combinatorics; integer partitions; finite set partitions; exchangeability

Special Issue Information

Dear Colleagues,

Loosely speaking, decomposable combinatorial structures are characterized via their components, such as cycles in permutations, parts in integer partitions, connected components in graphs, irreducible factors in polynomials over finite fields, etc. Such structures are ubiquitous in numerous areas of mathematics and its applications—from number theory, algebra, and topology to quantum physics, statistics, population genetics, IT, and cryptography.

Modern approach to “big” combinatorial structures is concerned with their typical macroscopic properties (e.g., limit shape) under a suitable probability measure. Of particular interest are possible effects caused by various constraints imposed, say, on the source of components, their multiplicities, total number, etc. Possible multivariate extensions include convex lattice chains (polygonal lines) and zonotopes. Furthermore, many topics in related domains may have a similar flavor—for example, in random matrix theory, with interesting subclasses exemplified by permutation matrices and band matrices.

This Special Issue aims at collecting high-quality papers to summarize and consolidate the state of the art in this fascinating research area along with modern advances in diverse applications.

Dr. Leonid V. Bogachev
Dr. Dirk Zeindler
Prof. Dr. Zhonggen Su
Dr. Yuri V. Yakubovich
Guest Editors

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Keywords

  • decomposable combinatorial structures
  • structural constraints
  • random integer partitions
  • finite set partitions
  • random permutations
  • spatial permutations
  • multivariate extensions (e.g., convex lattice chains, digital polyominoes, polytopes, zonotopes)
  • limit shapes
  • asymptotic fluctuations
  • generating functions
  • random matrix theory
  • applications (e.g., quantum physics, statistics, economics, population genetics, IT, cryptography)

Published Papers (6 papers)

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Research

15 pages, 278 KiB  
Article
Method for Obtaining Coefficients of Powers of Multivariate Generating Functions
by Dmitry Kruchinin, Vladimir Kruchinin and Yuriy Shablya
Mathematics 2023, 11(13), 2859; https://doi.org/10.3390/math11132859 - 26 Jun 2023
Cited by 2 | Viewed by 724
Abstract
There are several general concepts that allow obtaining explicit formulas for the coefficients of generating functions in one variable by using their powers. One such concept is the application of compositae of generating functions. In previous studies, we have introduced a generalization for [...] Read more.
There are several general concepts that allow obtaining explicit formulas for the coefficients of generating functions in one variable by using their powers. One such concept is the application of compositae of generating functions. In previous studies, we have introduced a generalization for the compositae of multivariate generating functions and have defined basic operations on the compositae of bivariate generating functions. The use of these operations helps to obtain explicit formulas for compositae and coefficients of generating functions in two variables. In this paper, we expand these operations on compositae to the case of generating functions in three variables. In addition, we describe a way of applying compositae to obtain coefficients of rational generating functions in several variables. To confirm the effectiveness of using the proposed method, we present detailed examples of its application in obtaining explicit formulas for the coefficients of a generating function related to the Aztec diamond and a generating function related to the permutations with cycles. Full article
(This article belongs to the Special Issue Random Combinatorial Structures)
30 pages, 448 KiB  
Article
Asymptotic Properties of Random Restricted Partitions
by Tiefeng Jiang and Ke Wang
Mathematics 2023, 11(2), 417; https://doi.org/10.3390/math11020417 - 12 Jan 2023
Viewed by 1180
Abstract
We study two types of probability measures on the set of integer partitions of n with at most m parts. The first one chooses the partition with a chance related to its largest part only. We obtain the limiting distributions of all of [...] Read more.
We study two types of probability measures on the set of integer partitions of n with at most m parts. The first one chooses the partition with a chance related to its largest part only. We obtain the limiting distributions of all of the parts together and that of the largest part as n tending to infinity for m fixed or tending to infinity with m=o(n1/3). In particular, if m goes to infinity not too fast, the largest part satisfies the central limit theorem. The second measure is very general and includes the Dirichlet and uniform distributions as special cases. The joint asymptotic distributions of the parts are derived by taking limits of n and m in the same manner as that in the first probability measure. Full article
(This article belongs to the Special Issue Random Combinatorial Structures)
23 pages, 430 KiB  
Article
Inverse Limit Shape Problem for Multiplicative Ensembles of Convex Lattice Polygonal Lines
by Leonid V. Bogachev and Sakhavet M. Zarbaliev
Mathematics 2023, 11(2), 385; https://doi.org/10.3390/math11020385 - 11 Jan 2023
Viewed by 1286
Abstract
Convex polygonal lines with vertices in Z+2 and endpoints at 0=(0,0) and n=(n1,n2), such that [...] Read more.
Convex polygonal lines with vertices in Z+2 and endpoints at 0=(0,0) and n=(n1,n2), such that n2/n1c(0,), under the scaling n11, have limit shape γ* with respect to the uniform distribution, identified as the parabola arc c(1x1)+x2=c. This limit shape is universal in a large class of so-called multiplicative ensembles of random polygonal lines. The present paper concerns the inverse problem of the limit shape. In contrast to the aforementioned universality of γ*, we demonstrate that, for any strictly convex C3-smooth arc γR+2 started at the origin and with the slope at each point not exceeding 90, there is a sequence of multiplicative probability measures Pnγ on the corresponding spaces of convex polygonal lines, under which the curve γ is the limit shape. Full article
(This article belongs to the Special Issue Random Combinatorial Structures)
21 pages, 364 KiB  
Article
A Measure-on-Graph-Valued Diffusion: A Particle System with Collisions and Its Applications
by Shuhei Mano
Mathematics 2022, 10(21), 4081; https://doi.org/10.3390/math10214081 - 2 Nov 2022
Viewed by 1294
Abstract
A diffusion-taking value in probability-measures on a graph with vertex set V, iVxiδi is studied. The masses on each vertex satisfy the stochastic differential equation of the form [...] Read more.
A diffusion-taking value in probability-measures on a graph with vertex set V, iVxiδi is studied. The masses on each vertex satisfy the stochastic differential equation of the form dxi=jN(i)xixjdBij on the simplex, where {Bij} are independent standard Brownian motions with skew symmetry, and N(i) is the neighbour of the vertex i. A dual Markov chain on integer partitions to the Markov semigroup associated with the diffusion is used to show that the support of an extremal stationary state of the adjoint semigroup is an independent set of the graph. We also investigate the diffusion with a linear drift, which gives a killing of the dual Markov chain on a finite integer lattice. The Markov chain is used to study the unique stationary state of the diffusion, which generalizes the Dirichlet distribution. Two applications of the diffusions are discussed: analysis of an algorithm to find an independent set of a graph, and a Bayesian graph selection based on computation of probability of a sample by using coupling from the past. Full article
(This article belongs to the Special Issue Random Combinatorial Structures)
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17 pages, 285 KiB  
Article
Generating Functions for Four Classes of Triple Binomial Sums
by Marta Na Chen and Wenchang Chu
Mathematics 2022, 10(21), 4025; https://doi.org/10.3390/math10214025 - 30 Oct 2022
Viewed by 884
Abstract
By means of the generating function approach, four classes of triple sums involving circular products of binomial coefficients are investigated. Recurrence relations and rational generating functions are established. Full article
(This article belongs to the Special Issue Random Combinatorial Structures)
22 pages, 402 KiB  
Article
On the Numbers of Particles in Cells in an Allocation Scheme Having an Even Number of Particles in Each Cell
by Alexey Nikolaevich Chuprunov and István Fazekas
Mathematics 2022, 10(7), 1099; https://doi.org/10.3390/math10071099 - 29 Mar 2022
Viewed by 1131
Abstract
We consider the usual random allocation model of distinguishable particles into distinct cells in the case when there are an even number of particles in each cell. For inhomogeneous allocations, we study the numbers of particles in the first K cells. We prove [...] Read more.
We consider the usual random allocation model of distinguishable particles into distinct cells in the case when there are an even number of particles in each cell. For inhomogeneous allocations, we study the numbers of particles in the first K cells. We prove that, under some conditions, this K-dimensional random vector with centralised and normalised coordinates converges in distribution to the K-dimensional standard Gaussian law. We obtain both local and integral versions of this limit theorem. The above limit theorem implies a χ2 limit theorem which leads to a χ2-test. The parity bit method does not detect even numbers of errors in binary files; therefore, our model can be applied to describe the distribution of errors in those files. For the homogeneous allocation model, we obtain a limit theorem when both the number of particles and the number of cells tend to infinity. In that case, we prove convergence to the finite dimensional distributions of the Brownian bridge. This result also implies a χ2-test. To handle the mathematical problem, we insert our model into the framework of Kolchin’s generalized allocation scheme. Full article
(This article belongs to the Special Issue Random Combinatorial Structures)
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