arXiv subjects

Markus Kuba

Publications and source records attributed to Markus Kuba.

At least 37 records · Page 2Linked to original sources

arXiv2021

On Path diagrams and Stirling permutations

A permutation can be locally classified according to the four local types: peaks, valleys, double rises and double falls. The corresponding classification of binary increasing trees uses four different types of nodes. Flajolet demonstrated the continued fraction representation of the generating function of local types, using a classical bijection between permutations, binary increasing trees, and suitably defined path diagrams induced by Motzkin paths. The aim of this article is to extend the notion of local types from permutations to $k$-Stirling permutations (also known as $k$-multipermutations). We establish a bijection of these local types to node types of $(k+1)$-ary increasing trees. We present a branched continued fraction representation of the generating function of these local types through a bijection with path diagrams induced by Łukasiewicz paths, generalizing the results from permutations to arbitrary $k$-Stirling permutations. We further show that the generating function of ordinary Stirling permutation has at least three branched continued fraction representations, using correspondences between non-standard increasing trees, $k$-Stirling permutations and path diagrams.

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arXiv2020

On bucket increasing trees, clustered increasing trees and increasing diamonds

In this work we analyze bucket increasing tree families. We introduce two simple stochastic growth processes, generating random bucket increasing trees of size $n$, complementing the earlier result of Mahmoud and Smythe for bucket recursive trees. On the combinatorial side, we define multilabelled generalizations of the tree families $d$-ary increasing trees and generalized plane-oriented recursive trees. Additionally, we introduce a clustering process for ordinary increasing trees and relate it to bucket increasing trees. We discuss in detail the bucket size two and present a bijection between such bucket increasing tree families and certain families of graphs called increasing diamonds, providing an explanation for phenomena observed by Bodini et al.

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arXiv2020

Logarithmic integrals, zeta values, and tiered binomial coefficients

We study logarithmic integrals of the form $\int_0^1 x^i\ln^n(x)\ln^m(1-x)dx$. They are expressed as a rational linear combination of certain rational numbers $(n,m)_i$, which we call tiered binomial coefficients, and products of the zeta values $ζ(2)$, $ζ(3)$,\dots. Various properties of the tiered binomial coefficients are established. They involve, amongst others, the binomial transform, truncated multiple zeta and multiple zeta star values, as well as special functions. As an application we discuss the limit law of the number of comparisons of the Quicksort algorithm: we reprove that the moments of the limit law are rational polynomials in the zeta values. A novel expression for the cumulants of the Quicksort limit is also presented.

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arXiv2018

A Note on the generating function of p-Bernoulli numbers

We use analytic combinatorics to give a direct proof of the closed formula for the generating function of $p$-Bernoulli numbers.

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arXiv2018

A Note on Harmonic number identities, Stirling series and multiple zeta values

We study a general type of series and relate special cases of it to Stirling series, infinite series discussed by Choi and Hoffman, and also to special values of the Arakawa-Kaneko zeta function, complementing and generalizing earlier results. Moreover, we survey properties of certain truncated multiple zeta and zeta star values, pointing out their relation to finite sums of harmonic numbers. We also discuss the duality result of Hoffman, relating binomial sums and truncated multiple zeta star values.

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arXiv2018

Combinatorial analysis of growth models for series-parallel networks

We give combinatorial descriptions of two stochastic growth models for series-parallel networks introduced by Hosam Mahmoud by encoding the growth process via recursive tree structures. Using decompositions of the tree structures and applying analytic combinatorics methods allows a study of quantities in the corresponding series-parallel networks. For both models we obtain limiting distribution results for the degree of the poles and the length of a random source-to-sink path, and furthermore we get asymptotic results for the expected number of source-to-sink paths. Moreover, we introduce generalizations of these stochastic models by encoding the growth process of the networks via further important increasing tree structures and give an analysis of some parameters.

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arXiv2018

An Asymptotic Series for an Integral

We obtain an asymptotic series $\sum_{j=0}^\infty\frac{I_j}{n^j}$ for the integral $\int_0^1[x^n+(1-x)^n]^{\frac1{n}}dx$ as $n\to\infty$, and compute $I_j$ in terms of alternating (or "colored") multiple zeta value. We also show that $I_j$ is a rational polynomial the ordinary zeta values, and give explicit formulas for $j\le 12$. As a byproduct, we obtain precise results about the convergence of norms of random variables and their moments. We study $\Vert(U,1-U)\Vert_n$ as $n$ tends to infinity and we also discuss $\Vert(U_1,U_2,\dots,U_r)\Vert_n$ for standard uniformly distributed random variables.

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arXiv2016

Classification of urn models with multiple drawings

We consider multicolor urn models with multiple drawings. An urn model is called linear if the conditional expected value of the urn composition at time $n$ is a linear function of the composition at time $n-1$. For four different sampling schemes - ordered and unordered samples with or without replacement - we classify urns into linear and non-linear models. We also discuss representations of the expected value and the covariance for linear models.

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arXiv2016

On martingale tail sums in affine two-color urn models with multiple drawings

In two recent works, Kuba and Mahmoud (arXiv:1503.090691 and arXiv:1509.09053) introduced the family of two-color affine balanced Polya urn schemes with multiple drawings. We show that, in large-index urns (urn index between $1/2$ and $1$) and triangular urns, the martingale tail sum for the number of balls of a given color admits both a Gaussian central limit theorem as well as a law of the iterated logarithm. The laws of the iterated logarithm are new even in the standard model when only one ball is drawn from the urn in each step (except for the classical Polya urn model). Finally, we prove that the martingale limits exhibit densities (bounded under suitable assumptions) and exponentially decaying tails. Applications are given in the context of node degrees in random linear recursive trees and random circuits.

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arXiv2016

Combinatorial analysis of growth models for series-parallel networks

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arXiv2015

2-Xor revisited: satisfiability and probabilities of functions

The problem 2-Xor-Sat asks for the probability that a random expression, built as a conjunction of clauses $x \oplus y$, is satisfiable. We revisit this classical problem by giving an alternative, explicit expression of this probability. We then consider a refinement of it, namely the probability that a random expression computes a specific Boolean function. The answers to both problems involve a description of 2-Xor expressions as multigraphs and use classical methods of analytic combinatorics by expressing probabilities through coefficients of generating functions.

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arXiv2015

Two-color balanced affine urn models with multiple drawings II: large-index and triangular urns

This is the second part of a two-part investigation. We continue the study of a class of balanced urn schemes on balls of two colors (white and black). At each drawing, a sample of size $m\ge 1$ is drawn from the urn and ball addition rules are applied; the special case $m=1$ of sampling only a single ball coincides with ordinary balanced urn models. We consider these multiple drawings under sampling with or without replacement. For the class of affine conditional expected value, we study the number of white balls after $n$ steps. The affine class is parametrized by $Λ$, specified by the ratio of the two eigenvalues of a reduced ball replacement matrix and the sample size, leading to three different cases: small-index urns ($Λ\le \frac 1 2$, and the case $Λ= \frac 1 2$ is critical), large-index urns ($Λ> \frac 1 2$), and triangular urns. In Part I we derived a central limit theorem for small index urns, and proved almost-sure convergence for large index and triangular urn models. In the present paper (Part II), we continue the study of affiance urn schemes and study the moments of large-index urns and triangular urn models. We show moment convergence under suitable scaling and we also provide expressions for the moments.

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arXiv2015

Two-colour balanced affine urn models with multiple drawings I: central limit theorems

This is a research endeavor in two parts. We study a class of balanced urn schemes on balls of two colours (say white and black). At each drawing, a sample of size $m\ge 1$ is drawn from the urn, and ball addition rules are applied. We consider these multiple drawings under sampling with or without replacement. We further classify ball addition matrices according to the structure of the expected value into affine and nonaffine classes. We give a necessary and sufficient condition for a scheme to be in the affine subclass. For the affine subclass, we get explicit results for the expected value and second moment of the number of white balls after $n$ steps and an asymptotic expansion of the variance. Moreover, we uncover a martingale structure, amenable to a central limit theorem formulation. This unifies several earlier works focused on special cases of urn models with multiple drawings. The class is parametrized by $Λ$, specified by the ratio of the two eigenvalues of a "reduced" ball replacement matrix and the sample size. We categorize the class into small-index urns ($Λ<\frac 1 2$), critical-index urns ($Λ= \frac 1 2$), and large-index urns ($Λ> \frac 1 2$), and triangular urns. In the present paper (Part I), we obtain central limit theorems for small- and critical-index urns and prove almost-sure convergence for triangular and large-index urns. In a companion paper (Part II), we discuss the moment structure of large-index urns and triangular urns.

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arXiv2014

Combinatorial families of multilabelled increasing trees and hook-length formulas

In this work we introduce and study various generalizations of the notion of increasingly labelled trees, where the label of a child node is always larger than the label of its parent node, to multilabelled tree families, where the nodes in the tree can get multiple labels. For all tree classes we show characterizations of suitable generating functions for the tree enumeration sequence via differential equations. Furthermore, for several combinatorial classes of multilabelled increasing tree families we present explicit enumeration results. We also present multilabelled increasing tree families of an elliptic nature, where the exponential generating function can be expressed in terms of the Weierstrass-p function or the lemniscate sine function. Furthermore, we show how to translate enumeration formulas for multilabelled increasing trees into hook-length formulas for trees and present a general "reverse engineering" method to discover hook-length formulas associated to such tree families.

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arXiv2014

On moment sequences and mixed Poisson distributions

In this article we survey properties of mixed Poisson distributions and probabilistic aspects of the Stirling transform: given a non-negative random variable $X$ with moment sequence $(μ_s)_{s\in\mathbb{N}}$ we determine a discrete random variable $Y$, whose moment sequence is given by the Stirling transform of the sequence $(μ_s)_{s\in\mathbb{N}}$, and identify the distribution as a mixed Poisson distribution. We discuss properties of this family of distributions and present a simple limit theorem based on expansions of factorial moments instead of power moments. Moreover, we present several examples of mixed Poisson distributions in the analysis of random discrete structures, unifying and extending earlier results. We also add several entirely new results: we analyse triangular urn models, where the initial configuration or the dimension of the urn is not fixed, but may depend on the discrete time $n$. We discuss the branching structure of plane recursive trees and its relation to table sizes in the Chinese restaurant process. Furthermore, we discuss root isolation procedures in Cayley-trees, a parameter in parking functions, zero contacts in lattice paths consisting of bridges, and a parameter related to cyclic points and trees in graphs of random mappings, all leading to mixed Poisson-Rayleigh distributions. Finally, we indicate how mixed Poisson distributions naturally arise in the critical composition scheme of Analytic Combinatorics.

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arXiv2013

Multiple isolation of nodes in recursive trees

We introduce the problem of isolating several nodes in random recursive trees by successively removing random edges, and study the number of random cuts that are necessary for the isolation. In particular, we analyze the number of random cuts required to isolate $\ell$ selected nodes in a size-n random recursive tree for three different selection rules, namely (i) isolating all of the nodes labelled 1,2,...,$\ell$ (thus nodes located close to the root of the tree), (ii) isolating all of the nodes labelled n+1-$\ell$,n+2-$\ell$,...,n (thus nodes located at the fringe of the tree), and (iii) isolating $\ell$ nodes in the tree, which are selected at random before starting the edge-removal procedure. Using a generating functions approach we determine for these selection rules the limiting distribution behavior of the number of cuts to isolate all selected nodes, for $\ell$ fixed and n tending to infinity.

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arXiv2011

On death processes and urn models

We use death processes and embeddings into continuous time in order to analyze several urn models with a diminishing content. In particular we discuss generalizations of the pill's problem, originally introduced by Knuth and McCarthy, and generalizations of the well known sampling without replacement urn models, and OK Corral urn models.

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arXiv2011

On generalized Pólya urn models

We study an urn model introduced in the paper of Chen and Wei, where at each discrete time step $m$ balls are drawn at random from the urn containing colors white and black. Balls are added to the urn according to the inspected colors, generalizing the well known Pólya-Eggenberger urn model, case m=1. We provide exact expressions for the expectation and the variance of the number of white balls after n draws, and determine the structure of higher moments. Moreover, we discuss extensions to more than two colors. Furthermore, we introduce and discuss a new urn model where the sampling of the m balls is carried out in a step-by-step fashion, and also introduce a generalized Friedman's urn model.

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