734 research outputs found
Analytic urns
This article describes a purely analytic approach to urn models of the
generalized or extended P\'olya-Eggenberger type, in the case of two types of
balls and constant ``balance,'' that is, constant row sum. The treatment starts
from a quasilinear first-order partial differential equation associated with a
combinatorial renormalization of the model and bases itself on elementary
conformal mapping arguments coupled with singularity analysis techniques.
Probabilistic consequences in the case of ``subtractive'' urns are new
representations for the probability distribution of the urn's composition at
any time n, structural information on the shape of moments of all orders,
estimates of the speed of convergence to the Gaussian limit and an explicit
determination of the associated large deviation function. In the general case,
analytic solutions involve Abelian integrals over the Fermat curve x^h+y^h=1.
Several urn models, including a classical one associated with balanced trees
(2-3 trees and fringe-balanced search trees) and related to a previous study of
Panholzer and Prodinger, as well as all urns of balance 1 or 2 and a sporadic
urn of balance 3, are shown to admit of explicit representations in terms of
Weierstra\ss elliptic functions: these elliptic models appear precisely to
correspond to regular tessellations of the Euclidean plane.Comment: Published at http://dx.doi.org/10.1214/009117905000000026 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
The shape of random tanglegrams
A tanglegram consists of two binary rooted trees with the same number of
leaves and a perfect matching between the leaves of the trees. We show that the
two halves of a random tanglegram essentially look like two independently
chosen random plane binary trees. This fact is used to derive a number of
results on the shape of random tanglegrams, including theorems on the number of
cherries and generally occurrences of subtrees, the root branches, the number
of automorphisms, and the height. For each of these, we obtain limiting
probabilities or distributions. Finally, we investigate the number of matched
cherries, for which the limiting distribution is identified as well
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.Comment: 37 page
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