Smolyak 网格中的节点计数

Counting Candy Crush Configurations

A k-stable c-coloured Candy Crush grid is a weak proper c-colouring of a particular type of k-uniform hypergraph. In this paper we introduce a fully polynomial randomised approximation scheme (FPRAS) which counts the number of k-stable c-coloured Candy Crush grids of a given size (m, n) for certain values of c and k. We implemented this algorithm on Matlab, and found that in a Candy Crush grid with7 available colours there are approximately 4.3*10^61 3-stable colourings. (Note that, typical Candy Crush games are played with 6 colours and our FPRAS is not guaranteed to work in expected polynomi…

Spanning Trees in Grid Graphs

Building on work by Desjarlais, Molina, Faase, and others, a general method is obtained for counting the number of spanning trees of graphs that are a product of an arbitrary graph and either a path or a cycle, of which grid graphs are a subclass. Results are obtained pertaining to recurrences obtained in this manner, and numerous new integer sequences are found.

Enumerating solutions to grid based puzzles with a fixed number of rows

In this paper we demonstrate a method for counting the number of solutions to various logic puzzles. Specifically, we remove all of the "clues" from the puzzle which help the solver to a unique solution, and instead start from an empty grid. We then count the number of ways to fill in this empty grid to a valid solution. We fix the number of rows $r$, vary the number of columns $k$, and then compute the sequence $A_r(k)$, which gives the number of solutions on an empty grid of size $r \times k$.

Intensive use of computing resources for dominations in grids and other combinatorial problems

Our goal is to prove new results in graph theory and combinatorics thanks to the speed of computers, used with smart algorithms. We tackle four problems. The four-colour theorem states that any map whose countries are connected can be coloured with 4 colours such that neighbouring countries have differnt colours. It was the first result proved using computers, in 1989. We wished to automatise further its proof. We explain the proof and provide a program which replays the proof. It also makes it possible to obtain other results with the same method. We give ideas to automatise the search for di…

几种无限图族的生成树自动计数

Automated Counting of Spanning Trees for Several Infinite Families of Graphs

本文通过实验数学方法，利用矩阵树定理和有理函数拟合，高效计算了几类无限图族的生成树数量。

任意网格图中连通集数量的一个紧上界

A sharp upper bound for the number of connected sets in any grid graph

本文通过多步递推公式建立了任意网格图中连通集数量的紧上界，推导出$(3\times n)$和$(4\times n)$网格图的枚举公式，并用转移矩阵法确定了$K_{m}\times P_{n}$的连通集数量，解决了Vince问题的一个特例。

Transcendence of generating functions of walks on the slit plane

Consider a single walker on the slit plane, that is, the square grid Z^2 without its negative x-axis, who starts at the origin and takes his steps from a given set S. Mireille Bousquet-Melou conjectured that -- excluding pathological cases -- the generating function counting the number of possible walks is algebraic if and only if the walker cannot cross the negative x-axis without touching it. In this paper we prove a special case of her conjecture.