Related papers: The Necklace Process: A Generating Function Approa…
Simple formulas for the number of different cyclic and dihedral necklaces containing $n_j$ beads of the $j$-th color, $j\leq m$ and $\sum_{j=1}^mn_j=N$, are derived.
We address two variants of the classical necklace counting problem from enumerative combinatorics. In both cases, we fix a finite group $\mathcal{G}$ and a positive integer $n$. In the first variant, we count the ``identity-product…
Bobbin lace is a fibre art form in which intricate and delicate patterns are created by braiding together many threads. An overview of how bobbin lace is made is presented and illustrated with a simple, traditional bookmark design. Research…
A necklace can be considered as a cyclic list of $n$ red and $n$ blue beads in an arbitrary order, and the goal is to fold it into two and find a large cross-free matching of pairs of beads of different colors. We give a counterexample for…
We define and analyze a coalescent process as a recursive box-filling process whose genealogy is given by an ancestral time-reversed, time-inhomogeneous Bienyam\'{e}-Galton-Watson process. Special interest is on the expected size of a…
Three-dimensional three-colour percolation on a lattice made of tetrahedra is a direct generalization of two-dimensional two-colour percolation on the triangular lattice. The interfaces between one-colour clusters are made of bicolour…
It can be conjectured that the colored Jones function of a knot can be computed in terms of counting paths on the graph of a planar projection of a knot. On the combinatorial level, the colored Jones function can be replaced by its weight…
Binary trees are fundamental objects in models of evolutionary biology and population genetics. Here, we discuss some of their combinatorial and structural properties as they depend on the tree class considered. Furthermore, the process by…
This survey article describes a method for choosing uniformly at random from any finite set whose objects can be viewed as constituting a distributive lattice. The method is based on ideas of the author and David Wilson for using ``coupling…
We present several multi-variable generating functions for a new pattern matching condition on the wreath product of the cyclic group and the symmetric group. Our new pattern matching condition requires that the underlying permutations…
We consider natural exponential families of Levy processes with randomized parameter. Such processes are Markov, and under suitable assumptions, pairs of such processes with shared randomization can be stitched together into a single…
Recent advances in classical density functional theory are combined with stochastic process theory and rare event techniques to formulate a theoretical description of nucleation, including crystallization, that can predict nonclassical…
We introduce bud generating systems, which are used for combinatorial generation. They specify sets of various kinds of combinatorial objects, called languages. They can emulate context-free grammars, regular tree grammars, and synchronous…
This article is a gentle introduction to the mathematical area known as circle packing, the study of the kinds of patterns that can be formed by configurations of non-overlapping circles. The first half of the article is an exposition of…
The necklace splitting problem is a classic problem in fair division with many applications, including data-informed fair hash maps. We extend necklace splitting to a dynamic setting, allowing for relocation, insertion, and deletion of…
We describe two general mechanisms for producing pairing bijections (bijective functions defined from N x N to N). The first mechanism, using n-adic valuations results in parameterized algorithms generating a countable family of distinct…
Combinatorial enumeration leads to counting generating functions presenting a wide variety of analytic types. Properties of generating functions at singularities encode valuable information regarding asymptotic counting and limit…
We propose and investigate a unifying class of sparse random graph models, based on a hidden coloring of edge-vertex incidences, extending an existing approach, Random graphs with a given degree distribution, in a way that admits a…
We study a system of interacting reinforced random walks defined on polygons. At each stage, each particle chooses an edge to traverse which is incident to its position. We allow the probability of choosing a given edge to depend on the sum…
We consider a Laplace operator on a random graph consisting of infinitely many loops joined symmetrically by intervals of unit length. The arc lengths of the loops are considered to be independent, identically distributed random variables.…