Bóna (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 Bóna. Here we will consider generalized Stirling permutations extending the earlier results of Bóna (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) Pólya urn models using various methods.
We study an urn model introduced in the paper of Chen and Wei (2005), 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.
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.
ABSTRACT. In two recent works, Mahmoud (arXiv:1503.090691 and arXiv:1509.09053) introduced the family of two-color affine balanced Pólya 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 Pólya 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.
ABSTRACT. We study a generalized Friedman's 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.
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