Lattice path
In combinatorics, a lattice path L in the d-dimensional integer lattice Z d {\displaystyle \mathbb {Z} ^{d}} of length k with steps in the set S, is a sequence of vectors v 0 , v 1 , … , v k ∈ Z d {\displaystyle v_{0},v_{1},\ldots ,v_{k}\in \mathbb {Z} ^{d}} such that each consecutive difference v i − v i − 1 {\displaystyle v_{i}-v_{i-1}} lies in S. A lattice path may lie in any lattice in R d {\displaystyle \mathbb {R} ^{d}} , but the integer lattice Z d {\displaystyle \mathbb {Z} ^{d}} is most commonly used. An example of a lattice path in Z 2 {\displaystyle \mathbb {Z} ^{2}} of length 5 with steps in S = { ( 2 , 0 ) , ( 1 , 1 ) , ( 0 , − 1 ) } {\displaystyle S=\lbrace (2,0),(1,1),(0,-1)\rbrace } is L = { ( − 1 , − 2 ) , ( 0 , − 1 ) , ( 2 , − 1 ) , ( 2 , − 2 ) , ( 2 , − 3 ) , ( 4 , − 3 ) } {\displaystyle L=\lbrace (-1,-2),(0,-1),(2,-1),(2,-2),(2,-3),(4,-3)\rbrace } .