47 research outputs found
Central Limit Theorems for some Set Partition Statistics
We prove the conjectured limiting normality for the number of crossings of a
uniformly chosen set partition of [n] = {1,2,...,n}. The arguments use a novel
stochastic representation and are also used to prove central limit theorems for
the dimension index and the number of levels
Part-products of -restricted integer compositions
If is a cofinite set of positive integers, an "-restricted composition
of " is a sequence of elements of , denoted
, whose sum is . For uniform random
-restricted compositions, the random variable is asymptotically lognormal. The proof is
based upon a combinatorial technique for decomposing a composition into a
sequence of smaller compositions.Comment: 18 page
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
Compositions into Powers of : Asymptotic Enumeration and Parameters
For a fixed integer base , we consider the number of compositions of
into a given number of powers of and, related, the maximum number of
representations a positive integer can have as an ordered sum of powers of .
We study the asymptotic growth of those numbers and give precise asymptotic
formulae for them, thereby improving on earlier results of Molteni. Our
approach uses generating functions, which we obtain from infinite transfer
matrices. With the same techniques the distribution of the largest denominator
and the number of distinct parts are investigated