Lattice path

In combinatorics, a lattice path ${\displaystyle L}$ in ${\displaystyle \mathbb {Z} ^{d}}$ of length ${\displaystyle k}$ with steps in ${\displaystyle S}$ is a sequence ${\displaystyle v_{0},v_{1},\ldots ,v_{k}\in \mathbb {Z} ^{d}}$ such that each consecutive difference ${\displaystyle v_{i}-v_{i-1}}$ lies in ${\displaystyle S}$. 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.

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