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 ${\displaystyle L=\lbrace (-1,-2),(0,-1),(2,-1),(2,-2),(2,-3),(4,-3)\rbrace }$.