Let \(P_x\),\(_y\)be the number of paths in \(S\) which end at x,y. Determine \(P_2\),\(_4\).
Details and assumptions
A lattice path is a path in the Cartesian plane between points with integer coordinates.
A step in a lattice path is a single move from one point with integer coordinates to another.
The size of the step from \((x_1\),\(y_1)\) to \((x_2\),\(y_2)\) is \((x_2 − x_1\),\(y_2 − y_1)\)
The length of a lattice path is the number of steps in the path.
For a set \(S =\) {(x\(_i\),y\(_i\))}\(^ k _{i=1}\) an \(S\)-lattice path is a lattice path where every step has size which is a member of \(S\).
