Special Lattice Paths

Let \(S\) be the set of \(\{(1,0), (0,1), (1,1), (1,-1), (-1,1)\}\)-lattice path which begin at \((1,1)\), do not use the same vertex twice, and never touch either the \(x\)-axis or the \(y\)-axis.

Let \(P_{x,y}\) be the number of paths in \(S\) which end at the point \((x,y)\). 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 \(S = \{(x_i,y_i)\}_{i=1}^{k}\), an \(S\)-lattice path is a lattice path where every step has size which is a member of \(S\).


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