Enumerating Up-Side Self-Avoiding Walks on Integer Lattices

Lauren K. Williams

Abstract


A self-avoiding walk (saw) is a path on a lattice that does not pass through the same point twice. Though mathematicians have studied saws for over fifty years, the number of $n$-step saws is unknown. This paper examines a special case of this problem, finding the number of $n$-step "up-side" saws (ussaws), saws restricted to moving up and sideways. It presents formulas for the number of $n$-step ussaws on various lattices, found using generating functions with decomposition and recursive methods.


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