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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]
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).
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 ...
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 ...
Since this set has both bilateral symmetry and rotational symmetry, it must have dihedral symmetry: the (n + 1)st dihedral group acts faithfully on this set. The size of this set is 2 n . For example, when n = 4, then the maximal element under the "staircase" that have rectangular Ferrers diagrams are
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.
A common type of lattice graph (known under different names, such as grid graph or square grid graph) is the graph whose vertices correspond to the points in the plane with integer coordinates, x-coordinates being in the range 1, ..., n, y-coordinates being in the range 1, ..., m, and two vertices being connected by an edge whenever the corresponding points are at distance 1.
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