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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]
The (large) Schröder numbers count both types of paths, and the little Schröder numbers count only the paths that only touch the diagonal but have no movements along it. [ 3 ] Just as there are (large) Schröder paths, a little Schröder path is a Schröder path that has no horizontal steps on the x {\displaystyle x} -axis.
An n-path from an n-tuple (,, …,) of vertices of G to an n-tuple (,, …,) of vertices of G will mean an n-tuple (,, …,) of paths in G, with each leading from to . This n -path will be called non-intersecting just in case the paths P i and P j have no two vertices in common (including endpoints) whenever i ≠ j {\displaystyle i\neq j} .
The Narayana numbers also count the number of lattice paths from (,) to (,), with steps only northeast and southeast, not straying below the x-axis, with peaks. The following figures represent the Narayana numbers N ( 4 , k ) {\displaystyle \operatorname {N} (4,k)} , illustrating the above mentioned symmetries.
Gauss's circle problem asks how many points there are inside this circle of the form (,) where and are both integers. Since the equation of this circle is given in Cartesian coordinates by x 2 + y 2 = r 2 {\displaystyle x^{2}+y^{2}=r^{2}} , the question is equivalently asking how many pairs of integers m and n there are such that
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63 Delannoy paths through a 3 × 3 grid. The octahedron in the three-dimensional integer lattice, whose number of lattice points is counted by the centered octahedral number, is a metric ball for three-dimensional taxicab geometry, a geometry in which distance is measured by the sum of the coordinatewise distances rather than by Euclidean distance.
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 ...