54,936 research outputs found
On the enumeration of leaf-labelled increasing trees with arbitrary node-degree
We consider the counting problem of the number of \textit{leaf-labeled
increasing trees}, where internal nodes may have an arbitrary number of
descendants. The set of all such trees is a discrete representation of the
genealogies obtained under certain population-genetical models such as
multiple-merger coalescents. While the combinatorics of the binary trees among
those are well understood, for the number of all trees only an approximate
asymptotic formula is known. In this work, we validate this formula up to
constant terms and compare the asymptotic behavior of the number of all
leaf-labelled increasing trees to that of binary, ternary and quaternary trees
Ordered increasing k-trees: Introduction and analysis of a preferential attachment network model
We introduce a random graph model based on k-trees, which can be generated by
applying a probabilistic preferential attachment rule, but which also has a
simple combinatorial description. We carry out a precise distributional
analysis of important parameters for the network model such as the degree, the
local clustering coefficient and the number of descendants of the nodes and
root-to-node distances. We do not only obtain results for random nodes, but in
particular we also get a precise description of the behaviour of parameters for
the j-th inserted node in a random k-tree of size n, where j = j(n) might grow
with n. The approach presented is not restricted to this specific k-tree model,
but can also be applied to other evolving k-tree models.Comment: 12 pages, 2 figure
On the genealogy of a population of biparental individuals
If one goes backward in time, the number of ancestors of an individual
doubles at each generation. This exponential growth very quickly exceeds the
population size, when this size is finite. As a consequence, the ancestors of a
given individual cannot be all different and most remote ancestors are repeated
many times in any genealogical tree. The statistical properties of these
repetitions in genealogical trees of individuals for a panmictic closed
population of constant size N can be calculated. We show that the distribution
of the repetitions of ancestors reaches a stationary shape after a small number
Gc ~ log N of generations in the past, that only about 80% of the ancestral
population belongs to the tree (due to coalescence of branches), and that two
trees for individuals in the same population become identical after Gc
generations have elapsed. Our analysis is easy to extend to the case of
exponentially growing population.Comment: 14 pages, 7 figures, to appear in the Journal of Theoretical Biolog
Efficiency of the Incomplete Enumeration algorithm for Monte-Carlo simulation of linear and branched polymers
We study the efficiency of the incomplete enumeration algorithm for linear
and branched polymers. There is a qualitative difference in the efficiency in
these two cases. The average time to generate an independent sample of
sites for large varies as for linear polymers, but as for branched (undirected and directed) polymers, where
. On the binary tree, our numerical studies for of order
gives . We argue that exactly in this
case.Comment: replaced with published versio
Bijective Enumeration of 3-Factorizations of an N-Cycle
This paper is dedicated to the factorizations of the symmetric group.
Introducing a new bijection for partitioned 3-cacti, we derive an el- egant
formula for the number of factorizations of a long cycle into a product of
three permutations. As the most salient aspect, our construction provides the
first purely combinatorial computation of this number
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