It is known that the moments of the maximum value of a one-dimensional conditional Brownian motion, the three-dimensional Bessel bridge with duration 1 started from the origin, are expressed using the Riemann zeta function. We consider a system of two Bessel bridges, in which noncolliding condition is imposed. We show that the moments of the maximum value is then expressed using the double Diri...

Classic results show that the Hawkes self-exciting point process can be viewed as a collection of temporal clusters, in which exogenously generated initial events give rise to endogenously driven descendant events. This perspective provides distribution cluster’s size through natural connection branching processes, but this is irrespective time. Insight into chronology cluster has been much mor...

We establish necessary and sufficient conditions for convergence (in the sense of finite dimensional distributions) of multiplicative measures on the set of partitions. We show that this convergence is equivalent to asymptotic independence of finite sizes of components. The multiplicative measures depict component spectra of random structures, the equilibrium of classic models of statistical me...

Coinductive counting: bisimulation in enumerative combinatorics (extended abstract) CWI's research has a theme-oriented structure and is grouped into four clusters. Listed below are the names of the clusters and in parentheses their acronyms. ABSTRACT The recently developed coinductive calculus of streams nds here a further application in enumer-ative c o m binatorics. A general methodology is ...

In this article, we solve a problem of enumerative combinatorics addressed and left open in [16]. The initial question is formulated in terms of paths on an infinite three-dimensional chessboard. The 3D chessboard being identified with N, a 3D rook is a piece which is allowed to move parallelly to one of the three axes. The general objective is to count paths (i.e., finite sequences of moves) o...

Let Hd(n, p) signify a random d-uniform hypergraph with n vertices in which each of the ( n d ) possible edges is present with probability p = p(n) independently, and let Hd(n, m) denote a uniformly distributed with n vertices and m edges. We derive local limit theorems for the joint distribution of the number of vertices and the number of edges in the largest component of Hd(n, p) and Hd(n, m)...

The class NP captures the difficulty of finding certificates. However, in many contexts, one is interested not just in a single certificate, but actually counting the number of certificates. In this manuscript we look at a famous result involving #P, (pronounced “sharp p”), a complexity class that captures this notion. Counting problems arise in diverse fields, often in situations having to do ...

In this paper, we focus on a “local property” of permutations: value-peak. A permutation σ has a value-peak σ(i) if σ(i − 1) < σ(i) > σ(i + 1) for some i ∈ [2, n − 1]. Define V P (σ) as the set of value-peaks of the permutation σ. For any S ⊆ [3, n], define V Pn(S) such that V P (σ) = S. Let Pn = {S | V Pn(S) 6= ∅}. we make the set Pn into a poset Pn by defining S T if S ⊆ T as sets. We prove t...

Given Schubert conditions on lines in projective space which generically determine a nite number of lines, we show there exist general real conditions determining the expected number of real lines. This extends the classical Schubert calculus of enu-merative geometry for the Grassmann variety of lines in projec-tive space from the complex realm to the real. Our main tool is an explicit geometri...

It is exponentially unlikely that a sparse random graph or hypergraph is connected, but such graphs occur commonly as the giant components of larger random graphs. This simple observation allows us to estimate the number of connected graphs, and more generally the number of connected d-uniform hypergraphs, on n vertices with m = O(n) edges. We also estimate the probability that a binomial rando...