Ljuben R. Mutafchiev

American University in Bulgaria · Mathematics and Science

Let Tn be the set of all mappings T : {1, 2,. .. , n} → {1, 2,. .. , n}. The corresponding graph of T is a union of disjoint connected unicyclic components. We assume that each T ∈ Tn is chosen uniformly at random (i.e., with probability n −n). The cycle of T contained within its largest component is called the deepest one. For any T ∈ Tn, let νn =...

Let $\mathcal{T}_n$ be the set of all mappings $T:\{1,2,\ldots,n\}\to\{1,2,\ldots,n\}$. The corresponding graph of $T$ is aunion of disjoint connected unicyclic components. We assume that each $T\in\mathcal{T}_n$ is chosen uniformly at random (i.e., with probability $n^{-n}$). The deepest cycle of $T$ is contained within its largest component. Let...

Пусть $p(n)$ - количество всех целочисленных разбиений положительного целого числа $n$, и пусть $\lambda $ - разбиение, выбранное случайно и равновероятно из всех таких $p(n)$ разбиений. Известно, что каждое разбиение $\lambda $ имеет единственное графическое представление, состоящее из $n$ неперекрывающихся ячеек на плоскости, называемое диаграммо...

Let $A$ be a set of natural numbers and let $S_{n,A}$ be the set of all permutations of $[n]=\{1,2,...,n\}$ with cycle lengths belonging to $A$. Furthermore, let $\mid A(n)\mid$ denote the cardinality of the set $A(n)=A\cap [n]$. The limit $\rho=\lim_{n\to\infty}\mid A(n)\mid/n$ (if it exists) is called the density of set $A$. It turns out that, as...

Let $A$ be a set of natural numbers and let $S_{n,A}$ be the set of all permutations of $[n]=\{1,2,...,n\}$ with cycle lengths belonging to $A$. For $A(n)=A\cap [n]$, the limit $\rho=\lim_{n\to\infty}\mid A(n)\mid/n$ (if it esists) is usually called the density of set $A$. (Here $\mid B\mid$ stands for the cardinality of the set $B$.) Several studi...

We propose an aproach for asymptotic analysis of plane partition statistics related to counts of parts whose sizes exceed a certain suitably chosen level. In our study, we use the concept of conjugate trace of a plane partition of the positive integer $n$, introduced by Stanley in 1973. We derive generating functions and determine the asymptotic be...

We study the asymptotic behavior of the maximal multiplicity $M_n=M_n(\sigma)$ of the block sizes in a set partition $\sigma$ of $[n]=\{1,2,....,n\}$, assuming that $\sigma$ is chosen uniformly at random from the set of all such partitions. It is known that, for large $n$, the blocks of a random set partition are typically of size $W=W(n)$, with $W...

We study the asymptotic behavior of the maximal multiplicity Mn = Mn(σ) of the blocks in a set partition of [n] = {1, 2, ..., n}, assuming that σ is chosen uniformly at random from the set of all such partitions. Let W = W (n) be the unique positive root of the equation W e W = n and let fn be the fractional part of W (n). Furthermore, let Rn = W W...

Let p(n) be the number of all integer partitions of the positive integer n and let λ be a partition, selected uniformly at random from among all such p(n) partitions. It is known that each partition λ has a unique graphical representation, composed by n non-overlapping cells in the plane called Young diagram. As a second step of our sampling experi...

Assuming that a plane partition of the positive integer n is chosen uniformly at random from the set of all such partitions, we propose a general asymptotic scheme for the computation of expectations of various plane partition statistics as n becomes large. The generating functions that arise in this study are of the form Q(x)F(x), where \(Q(x)=\pr...

Let $\lambda$ be a partition of the positive integer $n$ chosen umiformly at random among all such partitions. Let $L_n=L_n(\lambda)$ and $M_n=M_n(\lambda)$ be the largest part size and its multiplicity, respectively. For large $n$, we focus on a comparison between the partition statistics $L_n$ and $L_n M_n$. In terms of convergence in distributio...

Assuming that a plane partition of the positive integer $n$ is chosen uniformly at random from the set of all such partitions, we propose a general asymptotic scheme for the computation of expectations of various plane partition statistics as $n$ becomes large. The generating functions that arise in this study are of the form $Q(x)F(x)$, where $Q(x...

Let $\mathcal{P}=\{p_1,p_2,...\}$ be the set of all odd primes arranged in increasing order. A Goldbach partition of the even integer $2k>4$ is a way of writing it as a sum of two primes from $\mathcal{P}$ without regard to order. Let $Q(2k)$ be the number of all Goldbach partitions of the number $2k$. Assume that $2k$ is selected uniformly at rand...

Let $\Sigma_{2n}$ be the set of all unordered partitions of the even integers from the interval $(4,2n]$, $n>2$, into two odd prime parts. We select a partition from the set $\Sigma_{2n}$ uniformly at random. Let $2G_n$ be the number partitioned by this selection. $2G_n$ is sometimes called a Goldbach number. We recently showed (Electron. J. Combin...

Let $\Sigma_{2n}$ be the set of all partitions of the even integers from the interval $(4,2n], n>2,$ into two odd prime parts. We select a partition from the set $\Sigma_{2n}$ uniformly at random. Let $2G_n$ be the number partitioned by this selection. $2G_n$ is sometimes called a Goldbach number. In [6] we showed that $G_n/n$ converges weakly to t...

Let $\Sigma_{2n}$ be the set of all partitions of the even integers from the interval $[4,2n], n>2,$ into two prime parts. We show that $\mid\Sigma_{2n}\mid\sim 2n^2/\log^2{n}$ as $n\to\infty$. We also assume that a partition is selected uniformly at random from the set $\Sigma_{2n}$. Let $2X_n\in [4,2n]$ be the size of this partition. We prove a l...

Let $\lambda$ be a partition of the positive integer $n$, selected uniformly at random among all such partitions. Corteel et al. (1999) proposed three different procedures of sampling parts of $\lambda$ at random. They obtained limiting distributions of the multiplicity $\mu_n=\mu_n(\lambda)$ of the randomly-chosen part as $n\to\infty$. The asympto...

Let $$\lambda $$λ be a partition of the positive integer $$n$$n, selected uniformly at random among all such partitions. Corteel et al. (Random Stuct Algorithm 14:185–197, 1999) proposed three different procedures of sampling parts of $$\lambda $$λ at random. They obtained limiting distributions of the multiplicity of the randomly chosen part as $$...

Recent work of Brlek \textit{et al.} gives a characterization of digitally convex polyominoes using combinatorics on words. From this work, we derive a combinatorial symbolic description of digitally convex polyominoes and use it to analyze their limit properties and build a uniform sampler. Experimentally, our sampler shows a limit shape for large...

Let c m,n be the number of weighted partitions of the positive integer n with exactly m parts, 1≤m≤n. For a given sequence b k , k≥1, of part type counts (weights), the bivariate generating function of the numbers c m,n is given by the infinite product ∏ k=1 ∞ (1-uz k ) -b k · Let D(s)=∑ k=1 ∞ b k k -s , s=σ+iy, be the Dirichlet generating series o...

For a given sequence of weights (non-negative numbers), we consider partitions of the positive integer n. Each n-partition is selected uniformly at random from the set of all such partitions. Under a classical scheme of assumptions on the weight sequence, which are due to Meinardus (1954), we show that the largest part in a random weighted partitio...

The family tree of a Galton-Watson branching process may contain N-ary subtrees, i.e. subtrees whose vertices have at least N ≥ 1 children. For family trees without infinite N-ary subtrees, we study how fast N-ary subtrees of height t disappear as t → ∞. 1 Introduction and Statement of the Results The family tree associated with a Bieneimé-Galton-W...

We associate with a Bienayme-Galton-Watson branching process a family tree rooted at the ancestor. For a positive integer N, define a complete N-ary tree to be the family tree of a deterministic branching process with offspring generating function s^N. We study the random variables V(N,n) and V(N) counting the number of disjoint complete N-ary subt...

The paper presents a discussion on the asymptotic formula for the number of plane partitions of a large positive integer.

We study the asymptotic behavior of the largest part size of a plane partition ! of the positive integer n, assuming that ! is chosen uniformly at random from the set of all such partitions. We prove that this characteristic, appropriately normalized, tends weakly, as n ! " , to a random variable having an extreme value probability distribution wit...

We study the asymptotic behavior of the maximal multiplicity μn = μn (λ) of the parts in a partition λ of the positive integer n, assuming that λ is chosen uniformly at random from the set of all such partitions. We prove that πμn /(6n)1/2 converges weakly to max j X j /j as n→∞, where X 1, X 2, … are independent and exponentially distributed rando...

We study the asymptotic behaviour of the trace (the sum of the diagonal parts) of a plane partition of the positive integer n, assuming that this parfition is chosen uniformly at random from the set of all such partitions.

We introduce the generalized zero-inflated allocation scheme of placing n labeled balls into N labeled cells. We study the asymptotic behavior of the number of empty cells when (n,N) belongs to the “right” and “left” domain of attraction. An application to the estimation of characteristics of agreement among a set of raters which independently clas...

We prove a local limit theorem for the length of the side of the Durfee square in a random partition of a positive integer n as n→∞. We rely our asymptotic analysis on the power series expansion of xm2∏j=1m(1−xj)−2 whose coefficient of xn equals the number of partitions of n in which the Durfee square is m2.

For a partition \(\), of a positive integer n chosen uniformly at random from the set of all such partitions, the kth excess is defined by \(\) if \(\). We prove a bivariate local limit theorem for \(\) as \(\). The whole range of possible values of k is studied. It turns out that ρ and ηk are asymptotically independent and both follow the doubly e...

Let Fn be the set of random mappings ' : f1; : : : ; ng ! f1; : : : ; ng (such that every mapping is equally likely). For x 2 f1; : : : ; ng the elements S k0 ' k (fxg) are called the predecessors of x. Let Nr denote the random variable which counts the number of points x 2 f1; : : : ; ng with exactly r predecessors. In this paper we identify the l...

Let Y s,n denote the number of part sizes ≧ s in a random and uniform partition of the positive integer n that are counted without multiplicity. For s = λ(6n)1/2/π + o(n 1/4), 0 ≦ λ n → ∞, we establish the weak convergence of Y s,n to a Gaussian distribution in the form of a central limit theorem. The mean and the standard deviation are also asym...

We study the asymptotic behaviour of the sth largest part Ls,n in a random partition of a positive integer n as n → ∞. The weak convergence of the distribution of Ls,n to the Gaussian distribution is established provided s is of order n1/2 and n → ∞. The work was supported by the Bulgarian Ministry of Education, Science, and Technologies, contract...

We consider four families of forests on n vertices: labeled and unlabeled forests containing rooted and unrooted trees, respectively. A forest is chosen uniformly from one of the given four families. The limiting distribution of the size of its largest tree is then studied as n→∞. Convergences to discrete distributions depending on 1/2- and 3/2-sta...

We investigate from probabilistic point of view the asymptotic behavior of the number of distinct component sizes in general classes of combinatorial structures of sizenasn→∞. Mild restrictions of admissibility type are imposed on the corresponding generating functions and asymptotic expressions of the mean and variance of that number are obtained....

The discrete probability distribution function Pr(Y = m) = \frac1m!\fracdm - 1 dzm - 1 (f (z)gm (z))|z = 0 ,m = 1,2,...,\Pr (Y = m) = \frac{1}{{m!}}\frac{{d^{m - 1} }}{{dz^{m - 1} }}(f (z)g^m (z))|_{z = 0} ,m = 1,2,..., and Pr(Y = 0) =f(0) define the class of the Lagrangian distributions iff andg are arbitrary probability generating functions of r...

A random mapping (T; q) of a finite set V = {1, 2, . . . , n} into itself assigns independently to each i is-an-element-of V its unique image j = T(i) is-an-element-of V with probability q for i = j and with probability 1-q/n-1 for j not-equal i. The purpose of the article is to determine the asymptotic behavior of the size of the largest connected...

Mapping patterns may be represented by unlabelled directed graphs in which each point has outdegree one. We consider the uniform probability measure on the set of all mapping patterns on n points and derive the limiting distribution of the size of the largest tree as n → ∞. It is also shown that the asymptotic results concerning the structure of tr...

A Local limit theorem for the distribution of the number of components in random labelled relational structures of size n (e.g., a type of random graphs on n vertices, random permutations of n elements, etc.) is proved as n→∞. The case when the corresponding exponential generating functions diverge at their radii of convergence is considered.

For a random mapping model with a single attracting center (Stepanov (1971)) we study the relationship between the sizes of the central tree, the adjacent points, and the free points. Joint, marginal and conditional distributions are shown to be of well-known Lagrangian type. Exact and asymptotic moment properties are investigated with the aid of R...

For a random mapping model with a single attracting center (Stepanov (1971)) we study the relationship between the sizes of the central tree, the adjacent points, and the free points. Joint, marginal and conditional distributions are shown to be of well-known Lagrangian type. Exact and asymptotic moment properties are investigated with the aid of R...

Mapping patterns may be represented by unlabelled directed graphs in which each point has out-degree one. Assuming uniform probability distribution on the set of all mapping patterns onn points, we obtain limit distributions of some characteristics associated with the graphs of mapping patterns (connected and disconnected), asn. In particular, we s...

We consider random single-valued mappings of an n-element set into itself. Using simple probabilistic facts, we show that the number of nodes in low strata and the number of cyclic nodes of the graphs of such mappings are identically distributed as n --> [infinity].

We consider the random vector T = (T(0), ⋯, T(n)) with independent identically distributed coordinates such that Pr{T(i) = j} = Pj, j = 0, 1, ⋯, n, Σj = 0 n Pj = 1. A realization of T can be viewed as a random graph GT with vertices {0, ⋯, n} and arcs {(0, T(0)), ⋯, (n, T(n))}. For each T we partition the vertex-set of GT into three disjoint groups...

We consider the random vector T = (T(0), ···, T(n)) with independent identically distributed coordinates such that Pr{T(i) = j} = P j , j = 0, 1, ···, n, Σ . A realization of T can be viewed as a random graph G T with vertices {0, ···, n} and arcs {(0, T(0)), ···, (n, T(n))}. For each T we partition the vertex-set of G T into three disjoint group...

The paper presents a random sampling procedure from a Unite population and its applications to random graphs and epidemic processes on random mappings.

We rederive Wright's formula [20] for the main term in the asymptotic expansion of the number of plane partitions of a large positive integer. Our approach is based on the saddle point method. We avoid its application step by step using instead a general result due to Hayman [8] for the coefficients of a wide class of generating series.