Related papers: Lattice Path Enumeration
A prototypical problem on which techniques for exact enumeration are tested and compared is the enumeration of self-avoiding walks. Here, we show an advance in the methodology of enumeration, making the process thousands or millions of…
Enumerating polygons on regular lattices is a classic problem in rigorous statistical mechanics. The goal of enumerating polygons on the square lattice via fermionic path integration was achieved using a free-fermion quadratic action in the…
This paper is simply a collection of process diagrams for further use and reference. These are diagrams about different approaches to research.
A polynomial time algorithm which detects all paths and cycles of all lengths in form of vertex pairs (start, finish).
In this paper, a survey about recent progress on problems solved using graph amalgamations is presented, along with some new results with complete proofs, and some related open problems.
We present a comprehensive survey of constructions of the real numbers (from either the rationals or the integers) in a unified fashion, thus providing an overview of most (if not all) known constructions ranging from the earliest attempts…
This work develops a methodical approach to counting of walks on cartesian products, biproducts, symmetric and exterior powers and bipowers, Schur operations, coverings and semicoverings of weighted graphs. For weight and root lattices of…
Let $a,b$ be fixed positive coprime integers. For a positive integer $g$, write $W_k(g)$ for the set of lattice paths from the startpoint $(0,0)$ to the endpoint $(ga,gb)$ with steps restricted to $\{(1,0), (0,1)\}$, having exactly $k$…
This is a structured compilation of some of my favourite open problems.
Various lattice path models are reviewed. The enumeration is done using generating functions. A few bijective considerations are woven in as well. The kernel method is often used. Computer algebra was an essential tool. Some results are…
We show how the Hamiltonian lattice loop representation can be cast straightforwardly in the path integral formalism. The procedure is general for any gauge theory. Here we present in detail the simplest case: pure compact QED. We also…
An updated review is presented of lattice investigations of colour confinement.
We show that weighted path orders are special instances of a variant of semantic path orders. Exploiting this fact, we introduce a generalization of weighted path orders that goes beyond the realm of simple termination. Experimental data…
Given a graph $G(V, E)$ and a positive integer $k$ ($k \geq 1$), a simple path on $k$ vertices is a sequence of $k$ vertices in which no vertex appears more than once and each consecutive pair of vertices in the sequence are connected by an…
We introduce a notion of Dyck paths with coloured ascents. For several ways of colouring, we establish bijections between sets of such paths and other combinatorial structures, such as non-crossing trees, dissections of a convex polygon,…
The purpose of these notes is to introduce some of the problems the enumeration of lattice walks is dedicated to and familiarize with some of the arguments they can be addressed with. We discuss the enumeration of lattice walks, their…
A bargraph is a self-avoiding lattice path with steps $U=(0,1)$, $H=(1,0)$ and $D=(0,-1)$ that starts at the origin and ends on the $x$-axis, and stays strictly above the $x$-axis everywhere except at the endpoints. Bargraphs have been…
This thesis deals with the enumerative study of combinatorial maps, and its application to the enumeration of other combinatorial objects. Combinatorial maps, or simply maps, form a rich combinatorial model. They have an intuitive and…
This is a survey on stated skein algebras and their representations.
We present some old and new results in the enumeration of random walks in one dimension, mostly developed in works of enumerative combinatorics. The relation between the trace of the $n$-th power of a tridiagonal matrix and the enumeration…