Lattice path

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 $S$ .^[1] A lattice path may lie in any lattice in ⁠ $\mathbb {R} ^{d}$ ⁠,^[1] 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 $L=\lbrace (-1,-2),(0,-1),(2,-1),(2,-2),(2,-3),(4,-3)\rbrace$ .

^ ^a ^b Stanley, Richard (2012). Enumerative Combinatorics, Volume 1 (2 ed.). Cambridge University Press. p. 21. ISBN 978-1-107-60262-5.

[EC1-1] Stanley, Richard (2012). Enumerative Combinatorics, Volume 1 (2 ed.). Cambridge University Press. p. 21. ISBN 978-1-107-60262-5.

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