Markus Kuba

Generalized Stirling permutations, families of increasing trees and urn models

Bona (2007) [6] studied the distribution of ascents, plateaux and descents in the class of Stirling permutations, introduced by Gessel and Stanley (1978) [13]. Recently, Janson (2008) [17] 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 (2007) [6] and Janson (2008) [17], 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.

On the degree distribution of the nodes in increasing trees

Simple families of increasing trees can be constructed from simply generated tree families, if one considers for every tree of size n all its increasing labellings, i.e., labellings of the nodes by distinct integers of the set {1,...,n} in such a way that each sequence of labels along any branch starting at the root is increasing. Three such tree families are of particular interest: recursive trees, plane-oriented recursive trees and binary increasing trees. We study the quantity degree of node j in a random tree of size n and give closed formulae for the probability distribution and all factorial moments for those subclass of tree families, which can be constructed via a tree evolution process. Furthermore limiting distribution results of this parameter are given, which completely characterize the phase change behavior depending on the growth of j compared to n.

Analysis of a generalized Friedman's urn with multiple drawings

We study a generalized Friedmans urn model with multiple drawings of white and blue balls. After a drawing, the replacement follows a policy of opposite reinforcement. We give the exact expected value and variance of the number of white balls after a number of draws, and determine the structure of the moments. Moreover, we obtain a strong law of large numbers, and a central limit theorem for the number of white balls. Interestingly, the central limit theorem is obtained combinatorially via the method of moments and probabilistically via martingales. We briefly discuss the merits of each approach. The connection to a few other related urn models is briefly sketched.

On quickselect, partial sorting and multiple quickselect

We present explicit solutions of a class of recurrences related to the Quickselect algorithm. Thus we are immediately able to solve recurrences arising at the partial sorting problem, which are contained in this class. Furthermore we show how the partial sorting problem is connected to the Multiple Quickselect algorithm and present a method for the calculation of solutions for a class of recurrences related to the Multiple Quickselect algorithm. Further an analysis of an algorithm for sorting a subarray A[r... r + p - 1], given the array A[1... n], is provided.

Limiting distributions for a class of diminishing urn models

In this work we analyze a class of 2 × 2 Pólya-Eggenberger urn models with ball replacement matrix and c = pa with . We determine limiting distributions by obtaining a precise recursive description of the moments of the considered random variables, which allows us to deduce asymptotic expansions of the moments. In particular, we obtain limiting distributions for the pills problem a = c = d = 1, originally proposed by Knuth and McCarthy. Furthermore, we also obtain limiting distributions for the well-known sampling without replacement urn, a = d = 1 and c = 0, and generalizations of it to arbitrary and c = 0. Moreover, we obtain a recursive description of the moment sequence for a generalized problem.

Analysis of statistics for generalized stirling permutations

In this work we give a study of generalizations of Stirling permutations, a restricted class of permutations of multisets introduced by Gessel and Stanley [15]. First we give several bijections between such generalized Stirling permutations and various families of increasing trees extending the known correspondences of [20, 21]. Then we consider several permutation statistics of interest for generalized Stirling permutations as the number of left-to-right minima, the number of left-to-right maxima, the number of blocks of specified sizes, the distance between occurrences of elements, and the number of inversions. For all these quantities we give a distributional study, where the established connections to increasing trees turn out to be very useful. To obtain the exact and limiting distribution results we use several techniques ranging from generating functions, connections to urn models, martingales and Steins method.

A combinatorial approach to the analysis of bucket recursive trees

In this work we provide a combinatorial analysis of bucket recursive trees, which have been introduced previously as a natural generalization of the growth model of recursive trees. Our analysis is based on the description of bucket recursive trees as a special instance of the so-called bucket increasing trees, which is a family of combinatorial objects introduced in this paper. Using this combinatorial description we obtain exact and limiting distribution results for the parameter depth of a specified element, descendants of a specified element and degree of a specified element.

On edge-weighted recursive trees and inversions in random permutations

We introduce random recursive trees, where deterministically weights are attached to the edges according to the labeling of the trees. We will give a bijection between recursive trees and permutations, which relates the arising edge-weights in recursive trees with inversions of the corresponding permutations. Using this bijection we obtain exact and limiting distribution results for the number of permutations of size n, where exactly m elements have j inversions. Furthermore we analyze the distribution of the sum of labels of the elements, which have exactly j inversions, where we can identify Dickmans infinitely divisible distribution as the limit law. Moreover we give a distributional analysis of weighted depths and weighted distances in edge-weighted recursive trees.

Bilabelled increasing trees and hook-length formulae

We introduce two different kinds of increasing bilabellings of trees, for which we provide enumeration formulae. One of the bilabelled tree families considered is enumerated by the reduced tangent numbers and is in bijection with a tree family introduced by Poupard [11]. Both increasing bilabellings naturally lead to hook-length formulae for trees and forests; in particular, one construction gives a combinatorial interpretation of a formula for labelled unordered forests obtained recently by Chen et al. [1].

On the distribution of distances between specified nodes in increasing trees

We study the quantity distance between nodejand nodenin a random tree of sizen chosen from a family of increasing trees. For those subclass of increasing tree families, which can be constructed via a tree evolution process, we give closed formulae for the probability distribution, the expectation and the variance. Furthermore we derive a distributional decomposition of the random variable considered and we show a central limit theorem of this quantity, for arbitrary labels 1@?j~. Such tree models are of particular interest in applications, e.g., the widely used models of recursive trees, plane-oriented recursive trees and binary increasing trees are special instances and are thus covered by our results.