Search results
Results from the WOW.Com Content Network
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.
Lattice proteins are highly simplified models of protein-like heteropolymer chains on lattice conformational space which are used to investigate protein folding. [1] Simplification in lattice proteins is twofold: each whole residue ( amino acid ) is modeled as a single "bead" or "point" of a finite set of types (usually only two), and each ...
The Delannoy number (,) also counts the global alignments of two sequences of lengths and , [2] the points in an m-dimensional integer lattice or cross polytope which are at most n steps from the origin, [3] and, in cellular automata, the cells in an m-dimensional von Neumann neighborhood of radius n.
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.
The closely related large Schröder numbers are equal to twice the Schröder–Hipparchus numbers, and may also be used to count several types of combinatorial objects including certain kinds of lattice paths, partitions of a rectangle into smaller rectangles by recursive slicing, and parenthesizations in which a pair of parentheses surrounding the whole sequence of elements is also allowed.
The following is a list of structure types which appear in the tables above. Regarding the number of atoms in the unit cell, structures in the rhombohedral lattice system have a rhombohedral primitive cell and have trigonal point symmetry but are also often also described in terms of an equivalent but nonprimitive hexagonal unit cell with three ...
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} .