We study the asymptotic behaviour of the trace (the sum of the diagonal parts) τn = τn(ω) of a plane partition ω of the positive integer n, assuming that ω is chosen uniformly at random from the set of all such partitions. We prove that (τn − c0n2/3)/c1n1/3 log1/2n converges weakly, as n → ∞, to the standard normal distribution, where c0 = ζ(2)/ [2ζ(3)]2/3, c1 = √(1/3/) [2ζ(3)]1/3 and ζ(s) = Σj=1∞j−s.
We study probability distributions on all possible complete matchings in a complete bipartite graph, where the vertices in
both sets admit a linear order. We define a family of distributions, and give its equivalent implicit and explicit (parametric)
description: it is characterized implicitly by a collection of interesting conditional independence statements, or explicitly
by the property that the distributions belonging to the family factorize into factors which depend on “local” properties of
the matching. We also calculate the number of free parameters in this family.
The inclusion-exclusion principle is one of the basic theorems in combinatorics. In this paper the inclusion-exclusion principle for IF-sets on generalized probability measures is studied. The basic theorems are proved.
Authors:Guy Louchard, Helmut Prodinger, and Mark Ward
This paper complements the analysis of Louchard and Prodinger [LP08] on the number of rounds in a coin-flipping selection algorithm that occurs in the presence of a demon. We precisely analyze a very different aspect
of the selection algorithm, using different methods of analysis. Specifically, we obtain precise descriptions of the distribution
and all moments of the number of participants ultimately selected during the execution of the algorithm. The selection algorithm is robust in at least two significant
ways. The presence of a demon allows for the precise analysis even when errors may occur between the rounds of the selection
process. (The analysis also handles the more traditional case, in which no demon is involved.) The selection algorithm can
also use either biased or unbiased coins.