Lattice path

In combinatorics, a lattice path L in the d-dimensional integer lattice ${\displaystyle \mathbb {Z} ^{d}}$ of length k with steps in the set S, is a sequence of vectors ${\displaystyle v_{0},v_{1},\ldots ,v_{k}\in \mathbb {Z} ^{d}}$ such that each consecutive difference ${\displaystyle v_{i}-v_{i-1}}$ lies in S. A lattice path may lie in any lattice in ${\displaystyle \mathbb {R} ^{d}}$, but the integer lattice ${\displaystyle \mathbb {Z} ^{d}}$ is most commonly used.

An example of a lattice path in ${\displaystyle \mathbb {Z} ^{2}}$ of length 5 with steps in ${\displaystyle S=\lbrace (2,0),(1,1),(0,-1)\rbrace }$ is ${\displaystyle L=\lbrace (-1,-2),(0,-1),(2,-1),(2,-2),(2,-3),(4,-3)\rbrace }$.