Lattice path

Lattice path of length 5 in ℤ² with S = { (2,0), (1,1), (0,-1) }.

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

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