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Lattice path of length 5 in ℤ 2 with S = { (2,0), (1,1), (0,-1) }.. In combinatorics, a lattice path L in the d-dimensional integer lattice of length k with steps in the set S, is a sequence of vectors ,, …, such that each consecutive difference lies in S. [1]
A totally ordered set is a distributive lattice. 21. A metric lattice is modular. [6] 22. A modular lattice is semi-modular. [7] 23. A projective lattice is modular. [8] 24. A projective lattice is geometric. (def) 25. A geometric lattice is semi-modular. [9] 26. A semi-modular lattice is atomic. [10] [disputed – discuss] 27. An atomic ...
The acyclicity of G is an essential assumption in the Lindström–Gessel–Viennot lemma; it guarantees (in reasonable situations) that the sums (,) are well-defined, and it advects into the proof (if G is not acyclic, then f might transform a self-intersection of a path into an intersection of two distinct paths, which breaks the argument ...
Similarly, the Schröder numbers count the number of ways to divide a rectangle into + smaller rectangles using cuts through points given inside the rectangle in general position, each cut intersecting one of the points and dividing only a single rectangle in two (i.e., the number of structurally-different guillotine partitions).
In mathematics, a self-avoiding walk (SAW) is a sequence of moves on a lattice (a lattice path) that does not visit the same point more than once. This is a special case of the graph theoretical notion of a path. A self-avoiding polygon (SAP) is a closed self-avoiding walk on a lattice. Very little is known rigorously about the self-avoiding ...
The poset Y is a lattice. The meet and join of two partitions are given by the intersection and the union of the corresponding Young diagrams. Because it is a lattice in which the meet and join operations are represented by intersections and unions, it is a distributive lattice.
Traditionally, a topological space consists of a set of points together with a topology, a system of subsets called open sets that with the operations of union (as join) and intersection (as meet) forms a lattice with certain properties. Specifically, the union of any family of open sets is again an open set, and the intersection of finitely ...
In geometry and group theory, a lattice in the real coordinate space is an infinite set of points in this space with the properties that coordinate-wise addition or subtraction of two points in the lattice produces another lattice point, that the lattice points are all separated by some minimum distance, and that every point in the space is within some maximum distance of a lattice point.