arXiv subjects

Dudley Stark

Publications and source records attributed to Dudley Stark.

13 recordsLinked to original sources

arXiv2026

Poisson approximations of the number of fixed points in random multiset permutations

For ordinary permutations on $n$ letters, the distribution of the number of fixed points of random permutations is well known to approach the Poisson$(1)$ distribution in total variation distance as $n\to\infty$ super-exponentially quickly. We use Stein's method to get related results for the number of fixed points of random permutations of multisets. Given a sequence of multisets on $n$ letters whose expected number of fixed points converges to a constant $c$, we must have $c\geq 1$ and the distribution of number of fixed points converges to the ${\rm Poisson}(c)$ distribution as $n\to\infty$. If $c>1$, then the rate of convergence in total variation distance can be as slow as $n^{-1/2}$.

math.CO↗

arXiv2025

A Chinese restaurant process for multiset permutations

Multisets are like sets, except that they can contain multiple copies of their elements. If there are $n_i$ copies of $i$, $1\leq i\leq t$, in multiset $M_t$, then there are $\binom{n_1+\cdots+n_t}{n_1,\ldots, n_t}$ possible permutations of $M_t$. Knuth showed how to factor any multiset permutation into cycles. For fixed $n_i$, $i\geq 1$, we show how to adapt the Chinese restaurant process, which generates random permutations on $n$ elements with weighting $\theta^{\# \, {\rm cycles}}$, $\theta>0$, sequentially for $n=1,2,\ldots$, to the multiset case, where we fix the $n_i$ and build permutations on $M_t$ sequentially for $t=1,2,\ldots$. The number of cycles of a multiset permutation chosen uniformly at random, i.e.~$\theta=1$, has distribution given by the sum of independent negative hypergeometric distributed random variables. For all $\theta>0$, and under the assumption that $n_i=O(1)$, we show a central limit theorem as $t\to\infty$ for the number of cycles.

math.PR↗

arXiv2023

Single-cell mutational burden distributions in birth-death processes

Genetic mutations are footprints of tumour growth. While mutation data in bulk samples has been used to infer evolutionary parameters hard to measure in vivo, the advent of single-cell data has led to strong interest in the mutational burden distribution (MBD) among tumour cells. We introduce dynamical matrices and recurrence relations to integrate this single-cell MBD with known statistics, and derive new analytical expressions. Surprisingly, we find that the shape of the MBD is driven by cell lineage-level stochasticity rather than by the distribution of mutations in each cell division.

q-bio.PE↗

arXiv2022

Random Permutations and Queues

Given a growth rule which sequentially constructs random permutations of increasing degree, the stochastic process version of the rencontre problem asks what is the limiting proportion of time that the permutation has no fixed points (singleton cycles). We show that the discrete-time Chinese Restaurant Process (CRP) does not exhibit this limit. We then consider the related embedding of the CRP in continuous time and thereby show that it does have this and other limits of the time averages. By this embedding the cycle structure of the permutation can be represented as a tandem of infinite-server queues. We use this connection to show how results from the queuing theory can be interpreted in terms of the evolution of the cycle counts of permutations.

math.PR↗

arXiv2016

Poisson approximation of counts of subgraphs in random intersection graphs

Random intersection graphs are characterized by three parameters: $n$, $m$ and $p$, where $n$ is the number of vertices, $m$ is the number of objects, and $p$ is the probability that a given object is associated with a given vertex. Two vertices in a random intersection graph are adjacent if and only if they have an associated object in common. When $m=\lfloor n^α\rfloor$ for constant $α$, we provide a condition, called {\em strictly $α$-balanced}, for the Poisson convergence of the number of induced copies of a fixed subgraph.

math.CO↗

arXiv2016

The probability of nonexistence of a subgraph in a moderately sparse random graph

We develop a general procedure that finds recursions for statistics counting isomorphic copies of a graph $G_0$ in the common random graph models ${\cal G}(n,m)$ and ${\cal G}(n,p)$. Our results apply when the average degrees of the random graphs are below the threshold at which each edge is included in a copy of $G_0$. This extends an argument given earlier by the second author for $G_0=K_3$ with a more restricted range of average degree. For all strictly balanced subgraphs $G_0$, our results gives much information on the distribution of the number of copies of $G_0$ that are not in large "clusters" of copies. The probability that a random graph in ${\cal G}(n,p)$ has no copies of $G_0$ is shown to be given asymptotically by the exponential of a power series in $n$ and $p$, over a fairly wide range of $p$. A corresponding result is also given for ${\cal G}(n,m)$, which gives an asymptotic formula for the number of graphs with $n$ vertices, $m$ edges and no copies of $G_0$, for the applicable range of $m$. An example is given, computing the asymptotic probability that a random graph has no triangles for $p=o(n^{-7/11})$ in ${\cal G}(n,p)$ and for $m=o(n^{15/11})$ in ${\cal G}(n,m)$, extending results of the second author.

math.CO↗

arXiv2015

Developments in the Khintchine-Meinardus probabilistic method for asymptotic enumeration

A theorem of Meinardus provides asymptotics of the number of weighted partitions under certain assumptions on associated ordinary and Dirichlet generating functions. The ordinary generating functions are closely related to Euler's generating function $\prod_{k=1}^\infty S(z^k)$ for partitions, where $S(z)=(1-z)^{-1}$. By applying a method due to Khintchine, we extend Meinardus' theorem to find the asymptotics of the coefficients of generating functions of the form $\prod_{k=1}^\infty S(a_kz^k)^{b_k}$ for sequences $a_k$, $b_k$ and general $S(z)$. We also reformulate the hypotheses of the theorem in terms of generating functions. This allows us to prove rigorously the asymptotics of Gentile statistics and to study the asymptotics of combinatorial objects with distinct components.

math.PR↗

arXiv2012

A Meinardus theorem with multiple singularities

Meinardus proved a general theorem about the asymptotics of the number of weighted partitions, when the Dirichlet generating function for weights has a single pole on the positive real axis. Continuing \cite{GSE}, we derive asymptotics for the numbers of three basic types of decomposable combinatorial structures (or, equivalently, ideal gas models in statistical mechanics) of size $n$, when their Dirichlet generating functions have multiple simple poles on the positive real axis. Examples to which our theorem applies include ones related to vector partitions and quantum field theory. Our asymptotic formula for the number of weighted partitions disproves the belief accepted in the physics literature that the main term in the asymptotics is determined by the rightmost pole.

math.PR↗

arXiv2007

Meinardus' theorem on weighted partitions: extensions and a probabilistic proof

We give a probalistic proof of the famous Meinardus' asymptotic formula for the number of weighted partitions with weakened one of the three Meinardus' conditions, and extend the resulting version of the theorem to other two classis types of decomposable combinatorial structures, which are called assemblies and selections. The results obtained are based on combining Meinardus' analytical approach with probabilistic method of Khitchine.

math.PR↗

arXiv2007

Asymptotic enumeration of 2-covers and line graphs

In this paper we find asymptotic enumerations for the number of line graphs on $n$-labelled vertices and for different types of related combinatorial objects called 2-covers. We find that the number of 2-covers, $s_n$, and proper 2-covers, $t_n$, on $[n]$ both have asymptotic growth $$ s_n\sim t_n\sim B_{2n}2^{-n}\exp(-\frac12\log(2n/\log n))= B_{2n}2^{-n}\sqrt{\frac{\log n}{2n}}, $$ where $B_{2n}$ is the $2n$th Bell number, while the number of restricted 2-covers, $u_n$, restricted, proper 2-covers on $[n]$, $v_n$, and line graphs $l_n$, all have growth $$ u_n\sim v_n\sim l_n\sim B_{2n}2^{-n}n^{-1/2}\exp(-[\frac12\log(2n/\log n)]^2). $$ In our proofs we use probabilistic arguments for the unrestricted types of 2-covers and and generating function methods for the restricted types of 2-covers and line graphs.

math.CO↗

arXiv2006

Asymptotics for incidence matrix classes

We define {\em incidence matrices} to be zero-one matrices with no zero rows or columns. A classification of incidence matrices is considered for which conditions of symmetry by transposition, having no repeated rows/columns, or identification by permutation of rows/columns are imposed. We find asymptotics and relationships for the number of matrices with $n$ ones in these classes as $n\to\infty$.

math.CO↗

arXiv2005

Asymptotic enumeration of incidence matrices

We discuss the problem of counting {\em incidence matrices}, i.e. zero-one matrices with no zero rows or columns. Using different approaches we give three different proofs for the leading asymptotics for the number of matrices with $n$ ones as $n\to\infty$. We also give refined results for the asymptotic number of $i\times j$ incidence matrices with $n$ ones.

math.CO↗

arXiv2005

Asymptotic enumeration and logical limit laws for expansive multisets and selections

Given a sequence of integers $a_j, j\ge 1,$ a multiset is a combinatorial object composed of unordered components, such that there are exactly $a_j$ one-component multisets of size $j.$ When $a_j\asymp j^{r-1} y^j$ for some $r>0$, $y\geq 1$, then the multiset is called {\em expansive}. Let $c_n$ be the number of multisets of total size $n$. Using a probabilistic approach, we prove for expansive multisets that $c_n/c_{n+1}\to 1$ and that $c_n/c_{n+1}<1$ for large enough $n$. This allows us to prove Monadic Second Order Limit Laws for expansive multisets. The above results are extended to a class of expansive multisets with oscillation. Moreover, under the condition $a_j=Kj^{r-1}y^j + O(y^{νj}),$ where $K>0$, $r>0$, $y>1$, $ν\in (0,1)$, we find an explicit asymptotic formula for $c_n$. In a similar way we study the asymptotic behavior of selections which are defined as multisets composed of components of distinct sizes.

math.CO↗