Saturday, March 23, 2013

Straight undip words as lattice walks

To construct a straight word via its lattice walk:

Construct a diagonal walk beginning at the origin, staying in the first quadrant, and returning to the origin. This will serve as the diagonal skeleton. There are no other rules for the diagonal skeleton: it can retrace itself, or move ahead in a straight line.

Now trace the diagonal skeleton using the normal orthogonal moves (u/d for up/down; n/p for out/back). Each diagonal move in the skeleton needs two orthogonal moves to trace it, and there are always two choices in how this is done. Choose a way that does not result in an immediate retrace or two successive moves in the same direction.

Upon returning to the origin, you have the lattice walk of a straight undip word.

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