相关论文: Enumeration of lozenge tilings of hexagons with cu…

Can you decide if there is a coincidence in the numbers counting two different combinatorial objects? For example, can you decide if two regions in $\mathbb{R}^3$ have the same number of domino tilings? There are two versions of the…

We study the enumeration of bargraphs with respect to some corner statistics. We find generating functions for the number of bargraphs that tracks the corner statistics of interest, the number of cells, and the number of columns. The…

In this thesis we will present and discuss various results pertaining to tiling problems and mathematical logic, specifically computability theory. We focus on Wang prototiles, as defined in [32]. We begin by studying Domino Problems, and…

Using a calculus of variations approach, we determine the shape of a typical plane partition in a large box (i.e., a plane partition chosen at random according to the uniform distribution on all plane partitions whose solid Young diagrams…

Suppose $P$ is a symmetric convex polygon in the plane. We give a polynomial time algorithm that decides if $P$ can tile the plane by transations at some level (not necessarily at level one; this is multiple tiling). The main technical…

Ordinary binary multiplication of natural numbers can be generalized in a non-trivial way to a ternary operation by considering discrete volumes of lattice hexagons. With this operation, a natural notion of `3-primality' -- primality with…

We introduce a new method for studying gap probabilities in a class of discrete determinantal point processes with double contour integral kernels. This class of point processes includes uniform measures of domino and lozenge tilings as…

We calculate the generating functions for the number of tilings of rectangles of various widths by the right tromino, the $L$ tetromino, and the $T$ tetromino. This allows us to place lower bounds on the entropy of tilings of the plane by…

To flatten a set partition (with apologies to Mathematica) means to form a permutation by erasing the dividers between its blocks. Of course, the result depends on how the blocks are listed. For the usual listing--increasing entries in each…

We discuss the hard-hexagon and hard-square problems, as well as the corresponding problem on the honeycomb lattice. The case when the activity is unity is of interest to combinatorialists, being the problem of counting binary matrices with…

We describe some theoretical results on triangulations of surfaces and we develop a theory on roots, decompositions and genus-surfaces. We apply this theory to describe an algorithm to list all triangulations of closed surfaces with at most…

Enumerative algebraic geometry deals with problems of counting geometric objects defined algebraically, An important class of enumerative problems is that of counting curves: given a class of curves in some projective variety defined by…

We develop the connection of Berg partitions with special substitution tilings of two tiles. We obtain a new proof that the number of Berg partitions with a fixed connectivity matrix is equal to half of the sum of its entries, \cite{S-W}.…

The exactly solvable four-vertex model with the fixed boundary conditions in the presence of inhomogeneous linearly growing external field is considered. The partition function of the model is calculated and represented in the determinantal…

Wang tiles enable efficient pattern compression while avoiding the periodicity in tile distribution via programmable matching rules. However, most research in Wang tilings has considered tiling the infinite plane. Motivated by emerging…

Finding, counting and/or listing triangles (three vertices with three edges) in large graphs are natural fundamental problems, which received recently much attention because of their importance in complex network analysis. We provide here a…

An equilateral triangle cannot be dissected into finitely many mutually incongruent equilateral triangles [Tutte 1948]. Therefore Tuza [Tuza 1991] asked for the largest number $s=s(n)$ such that there is a tiling of an equilateral triangle…

The partitioning of space by hyperplanes in the context of discrete classification problem is considered. We obtain some relations for the number of partitions and establish a recurrence relation for the maximal number of partitions of R^n…

Enumeration algorithms have been one of recent hot topics in theoretical computer science. Different from other problems, enumeration has many interesting aspects, such as the computation time can be shorter than the total output size, by…

We consider the problem of optimizing the product of the distances from a given point in a triangle to each vertex. There are two possible cases in general. For isosceles triangles, we explicitly show exactly when both cases occur.