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Ljuben Mutafchiev

Publications and source records attributed to Ljuben Mutafchiev.

At least 19 recordsLinked to original sources
arXiv2025

Large Components and Trees of Random Mappings

Let $\mathcal{T}_n$ be the set of all mappings $T:[n]\to[n]$, where $[n]=\{1,2,\ldots,n\}$. The corresponding graph $G_T$ of $T$, called a functional digraph, is a union of disjoint connected components. Each component is a directed cycle of rooted labeled trees. We assume that each $T\in\mathcal{T}_n$ is chosen uniformly at random from the set $\mathcal{T}_n$. The components and trees of $G_T$ are distinguished by their size. In this paper, we compute the limiting conditional probability ($n\to\infty$) that a vertex from the largest component of the random graph $G_T$, chosen uniformly at random from $[n]$, belongs to its $s$-th largest tree, where $s\ge 1$ is a fixed integer. This limit can be also viewed as an approximation of the probability that the $s$-th largest tree of $G_T$ is a subgraph of its largest component, which is a solution of a problem suggested by Mutafchiev and Finch (2024).

math.CO↗
arXiv2025

A note on the distribution of the sum of lengths of the initial longest increasing sequences in cycles of random permutations

Let $S_n$ be the set of all permutations of $\{1,2,\ldots,n\}$ and let $\sigma=(\sigma_1,\sigma_2,\ldots,\sigma_n)\in S_n$. The {\it initial longest increasing sequence} (ILIS) in $\sigma$ has length $m$ if, for $1\le m\le n-1$, $\sigma_1<\sigma_2<\ldots<\sigma_m, \sigma_m>\sigma_{m+1}$, and has length $n$ if $\sigma=(1,2,\ldots,n)$. Let $l(\sigma)$ be the length of the ILIS in $\sigma$. We assume that $\sigma$ is represented in cycle notation, so that the first number in each cycle is the minimum number of this cycle. We also assume that $\sigma$ is chosen uniformly at random from $S_n$, i.e., with probability $1/n!$. Let $C_n(\sigma)$ be the set of all cycles of $\sigma$. In [9], T. Mansour investigated enumerative properties related to lengths of the ILIS in random permutations represented by the cycle notation. In particular, he studied the sum of the ILIS' lengths defined by $s_n=\sum_{c\in C_n(\sigma)} l(c)$ and derived exact and asymptotic expressions for its expectation and variance. In this note, we supplement Mansour's results on $s_n$ with a limit theorem. We show that $s_n$, appropriately normalized, converges weakly to a standard normal random variable as $n\to\infty$.

math.CO↗
arXiv2023

On the Deepest Cycle of a Random Mapping

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 a union 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 cycle of $T$ contained within its largest component is callled the deepest one. For any $T\in\mathcal{T}_n$, let $\nu_n=\nu_n(T)$ denote the length of this cycle. In this paper, we establish the convergence in distribution of $\nu_n/\sqrt{n}$ and find the limits of its expectation and variance as $n\to\infty$. For $n$ large enough, we also show that nearly $55\%$ of all cyclic vertices of a random mapping $T\in\mathcal{T}_n$ lie in the deepest cycle and that a vertex from the longest cycle of $T$ does not belong to its largest component with approximate probability $0.075$.

math.CO↗
arXiv2022

The asymptotic of the number of permutations whose cycle lengths are prime numbers

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 $n\to\infty$, the cardinality $\mid S_{n,A}\mid$ of the set $S_{n,A}$ essentially depends on $\rho$. The case $\rho>0$ was studied by several authors under certain additional conditions on $A$. In 1999, Kolchin noticed that there is a lack studies on classes of permutations for which $\rho=0$. In this context, he also proposed investigations on certain particular cases. In this paper, we consider the permutations whose cycle lengths are prime numbers, that is, we assume that $A=\mathcal{P}$, where $\mathcal{P}$ denotes the set of all primes. For this class of permutations, the Prime Number Theorem implies that $\rho=0$. In this paper, we show that, as $n\to\infty$, the ratio $S_{n,\mathcal{P}}/(n-1)!$ approaches a finite limit and determine its value explicitly. Our method of proof employs classical Tauberian theorems.

math.CO↗
arXiv2021

A Note on the Number of Permutations whose Cycle Lengths Are Prime Numbers

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 studies show that the asymptotic behavior of the cardinality $\mid S_{n,A}\mid$, as $n\to\infty$, depends on the density $\rho$. It turns out that the asumption $\rho>0$ plays an essential role in the asymptotic analysis of $\mid S_{n,A}\mid$. Kolchin (1999) noticed that there is a lack of studies on classes of permutations satisfying $\rho=0$ and proposed investigations on certain particular cases. In this note, we consider the permutations whose cycle lengths are prime numbers, that is, we assume that $A=\mathcal{P}$, where $\mathcal{P}$ denotes the set of all primes. From the Prime Number Theorem it follows that $\rho=0$ for this class of permutations. We deduce an asymptotic formula for the summatory function $\sum_{k\le n}\mid S_{k,\mathcal{P}}\mid/k!$ as $n\to\infty$. In our proof we employ the classical Hardy-Littlewood-Karamata Tauberian theorem.

math.CO↗
arXiv2020

Large Parts of Random Plane Partitions: a Poisson Limit Theorem

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 behavior of counts of large parts using a general scheme based on the saddle point method. In this way, we are able to prove a Poisson limit theorem for the number of parts of a random and uniformly chosen plane partition of $n$, whose sizes are greater than a function $m=m(n)$ as $n\to\infty$. An explicit expression for $m(n)$ is also given.

math.CO↗
arXiv2019

The limiting distribution of the hook length of a randomly chosen cell in a random Young diagram

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 experiment, we select a cell $c$ uniformly at random from among all $n$ cells of the Young diagram of the partition $λ$. For large $n$, we study the asymptotic behavior of the hook length $Z_n=Z_n(λ,c)$ of the cell $c$ of a random partituion $λ$. This two-step sampling procedure suggests a product probability measure, which assigns the probability $1/np(n)$ to each pair $(λ,c)$. With respect to this probability measure, we show that the random variable $πZ_n/\sqrt{6n}$ converges weakly, as $n\to\infty$, to a random variable whose probability density function equals $6y/π^2 (e^y-1)$ if $0<y<\infty$, and zero elsewhere.

math.CO↗
arXiv2019

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

We study the asymptotic behavior of the maximal multiplicity $M_n=M_n(σ)$ 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 $We^W=n$ and let $f_n$ be the fractional part of $W(n)$. Furthermore, let $R_n=W^{\lfloor W\rfloor}/\lfloor W\rfloor !$ and let $\vartheta_n=\min{\{f_n,1-f_n\}}$. We show that, over a subsequence $\{n_k\}_{k\ge 1}$, $(M_{n_k}-R_{n_k})/\sqrt{R_{n_k}}$ converges weakly, as $k\to\infty$, to $\max{\{Z_1,Z_2-u\}}$, where $Z_1$ and $Z_2$ are two independent copies of a standard normal random variable and either $u=\left(\frac{1}{2π}\right)^{1/4}\lim_{k\to\infty}\vartheta_{n_k}\frac{\sqrt{n_k}}{\log^{7/4}{n_k}}\in [0,\infty)$ or $u=\infty$. The proof uses the saddle point method. A comparison with the similar statistic for random integer partitions of $n$ is also given.

math.CO↗
arXiv2017

On the Largest Part Size and Its Multiplicity of a Random Integer Partition

Let $λ$ be a partition of the positive integer $n$ chosen umiformly at random among all such partitions. Let $L_n=L_n(λ)$ and $M_n=M_n(λ)$ 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 distribution, we show that they behave in the same way. However, it turns out that the expectation of $L_n M_n -L_n$ grows as fast as $\frac{1}{2}\log{n}$ We obtain a precise asymptotic expansion for this expectation and conclude with an open problem arising from this study.

math.PR↗
arXiv2017

Asymptotic Analysis of Expectations of Plane Partition Statistics

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 $Q(x)=\prod_{j=1}^\infty (1-x^j)^{-j}$ is the generating function for the number of plane partitions. We show how asymptotics of such expectations can be obtained directly from the asymptotic expansion of the function $F(x)$ around $x=1$. The representation of a plane partition as a solid diagram of volume $n$ allows interpretations of these statistics in terms of its dimensions and shape. As an application of our main result, we obtain the asymptotic behavior of the expected values of the largest part (the height of the solid diagram) and the trace (the number of cubes in the wall on the main diagonal of the solid diagram). Our results are similar to those of Grabner et al. (2014) related to linear integer partition statistics. We base our study on Hayman's method for admissible power series.

math.CO↗
arXiv2016

On the Distribution of the Number of Goldbach Partitions of a Randomly Chosen Positive Even Integer

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 random from the interval $(4,2n], n>2$, and let $Y_n=Q(2k)$ with probability $1/(n-2)$. We prove that the random variable $\frac{Y_n}{n/\left(\frac{1}{2}\log{n}\right)^2}$ converges weakly, as $n\to\infty$, to a uniformly distributed random variable in the interval $(0,1)$. The method of proof uses size-biasing and the Laplace transform continuity theorem.

math.PR↗
arXiv2015

Sampling Goldbach Numbers at Random

Let $Σ_{2n}$ 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 $Σ_{2n}$ 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 the maximum $T$ of two random variables which are independent copies of a uniformly distributed random variable in the interval $(0,1)$. In this note we show that the mean and the variance of $G_n/n$ tend to the mean $μ_T=2/3$ and variance $σ_T^2=1/18$ of $T$, respectively. Our method of proof is based on generating functions and on a Tauberian theorem due to Hardy-Littlewood-Karamata.

math.NT↗
arXiv2015

A Note on Goldbach Partitions of Large Even Integers

Let $Σ_{2n}$ be the set of all partitions of the even integers from the interval $(4,2n], n>2,$ into two odd prime parts. We show that $\midΣ_{2n}\mid\sim 2n^2/\log^2{n}$ as $n\to\infty$. We also assume that a partition is selected uniformly at random from the set $Σ_{2n}$. Let $2X_n\in (4,2n]$ be the size of this partition. We prove a limit theorem which establishes that $X_n/n$ converges weakly to the maximum of two random variables which are independent copies of a uniformly distributed random variable in the interval $(0,1)$. Our method of proof is based on a classical Tauberian theorem due to Hardy, Littlewood and Karamata. We also show that the same asymptotic approach can be applied to partitions of integers into an arbitrary and fixed number of odd prime parts

math.NT↗
arXiv2014

Sampling Parts of Random Integer Partitions: A Probabilistic and Asymptotic Analysis

Let $λ$ 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 $λ$ at random. They obtained limiting distributions of the multiplicity $μ_n=μ_n(λ)$ of the randomly-chosen part as $n\to\infty$. The asymptotic behavior of the part size $σ_n=σ_n(λ)$, under these sampling conditions, was found by Fristedt (1993) and Mutafchiev (2014). All these results motivated us to study the relationship between the size and the multiplicity of a randomly-selected part of a random partition. We describe it obtaining the joint limiting distributions of $(μ_n,σ_n)$, as $n\to\infty$, for all these three sampling procedures. It turns out that different sampling plans lead to different limiting distributions for $(μ_n,σ_n)$. Our results generalize those obtained earlier and confirm the known expressions for the marginal limiting distributions of $μ_n$ and $σ_n$.

math.PR↗
arXiv2014

Sampling Part Sizes of Random Integer Partitions

We consider procedures of sampling parts from a random integer partition. We determine asymptotically the probabilty distribution of the randomly-selected part whenever the positive integer that is partitioned becomes large.

math.PR↗
arXiv2013

The Size of the Largest Part of Random Weighted Partitions of Large Integers

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 partition, appropriately normalized, converges weakly, as n tends to infinity, to a random variable having the extreme value (Gumbel's) distribution. This limit theorem extends some known results on particular types of integer partitions and on the Bose-Einstein model of ideal gas.

math.PR↗