# 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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