arXiv subjects

Markus Kuba

Publications and source records attributed to Markus Kuba.

43 records · Page 3Linked to original sources

arXiv2010

On Sampling without replacement and OK-Corral urn models

In this work we discuss two urn models with general weight sequences $(A,B)$ associated to them, $A=(α_n)_{n\in\N}$ and $B=(β_m)_{m\in\N}$, generalizing two well known Pólya-Eggenberger urn models, namely the so-called sampling without replacement urn model and the OK Corral urn model. We derive simple explicit expressions for the distribution of the number of white balls, when all black have been drawn, and obtain as a byproduct the corresponding results for the Pólya-Eggenberger urn models. Moreover, we show that the sampling without replacement urn models and the OK Corral urn models with general weights are dual to each other in a certain sense. We discuss extensions to higher dimensional sampling without replacement and OK Corral urn models, respectively, where we also obtain explicit results for the probability mass functions, and also an analog of the dualitiy relation. Finally, we derive limit laws for a special choice of the weight sequences.

math.CO↗

arXiv2010

A note on hook length formulas for trees

In this short note we discuss recent results on hook length formulas of trees unifying some earlier results, and explain hook length formulas naturally associated to families of increasingly labelled trees.

math.CO↗

arXiv2009

On embedded trees and lattice paths

Bouttier, Di Francesco and Guitter introduced a method for solving certain classes of algebraic recurrence relations arising the context of embedded trees and map enumeration. The aim of this note is to apply this method to three problems. First, we discuss a general family of embedded binary trees, trying to unify and summarize several enumeration results for binary tree families, and also to add new results. Second, we discuss the family of embedded $d$-ary trees, embedded in the plane in a natural way. Third, we show that several enumeration problems concerning simple families of lattice paths can be solved without using the kernel method by regarding simple families of lattice paths as degenerated families of embedded trees.

math.CO↗

arXiv2009

On a reciprocity law for finite multiple zeta values

It was shown in that harmonic numbers satisfy certain reciprocity relations, which are in particular useful for the analysis of the quickselect algorithm. The aim of this work is to show that a reciprocity relation from \cite{KirProd98,ProSchnKu} can be generalized to finite variants of multiple zeta values, involving a finite variant of the shuffle identity for multiple zeta values. We present the generalized reciprocity relation and furthermore a simple elementary proof of the shuffle identity using only partial fraction decomposition. We also present an extension of the reciprocity relation to weighted sums.

math.CO↗

arXiv2009

On functions of Arakawa and Kaneko and multiple zeta functions

We study for $s\in\N=\{1,2,...\}$ the functions $ξ_{k}(s)=\frac{1}{Γ(s)}\int_{0}^{\infty}\frac{t^{s-1}}{e^t-1}\Li_{k}(1-e^{-t})dt$, and more generally $ξ_{k_1,...,k_r}(s)=\frac{1}{Γ(s)}\int_{0}^{\infty}\frac{t^{s-1}}{e^t-1}\Li_{k_1,...,k_r}(1-e^{-t})dt$, introduced by Arakawa and Kaneko \cite{Arakawa} and relate them with (finite) multiple zeta functions, partially answering a question of \cite{Arakawa}. In particular, we give an alternative proof of a result of Ohno \cite{Ohno2}.

math.NT↗

arXiv2009

A note on naturally embedded ternary trees

In this note we consider ternary trees naturally embedded in the plane in a deterministic way such that the root has position zero, or in other words label zero, and the children of a node with position $j$ have positions $j-1$, $j$, and $j+1$, for all $j\in\Z$. We derive the generating function of ternary trees where all nodes have labels which are less or equal than $j$, with $j\in\N$, and the generating function of ternary trees counted with respect to nodes with label $j$, with $j\in\Z$. Moreover, we discuss generalizations of the counting problem to several labels at the same time. Furthermore, we use generating functions to study the depths of the external node $s$, or in other words leaf $s$ with $0\le s\le 2n$, where the $2n+1$ external nodes of a ternary tree are numbered from the left to the right according to an inorder traveral. The three different types depths -- left, right and center -- are due to the embedding of the ternary tree in the plane. Finally, we discuss generalizations of the considered enumeration problems to embedded $d$-ary trees.

math.CO↗

arXiv2008

Generalized Stirling permutations, families of increasing trees and urn models

Bona [2007+] studied the distribution of ascents, plateaux and descents in the class of Stirling permutations, introduced by Gessel and Stanley [1978]. Recently, Janson [2008+] showed the connection between Stirling permutations and plane recursive trees and proved a joint normal law for the parameters considered by Bona. Here we will consider generalized Stirling permutations extending the earlier results of Bona and Janson, and relate them with certain families of generalized plane recursive trees, and also $(k+1)$-ary increasing trees. We also give two different bijections between certain families of increasing trees, which both give as a special case a bijection between ternary increasing trees and plane recursive trees. In order to describe the (asymptotic) behaviour of the parameters of interests, we study three (generalized) Polya urn models using various methods.

math.CO↗