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Jul 31, 2011 at 17:03 comment added Derrick Stolee That should be $r!/2^r$.
Jul 31, 2011 at 17:03 comment added Derrick Stolee If we assume the path moves in each dimension 0 or 2 times, you can select ${n \choose r/2}$ dimensions and then permute them $r^1/2^r$ ways. This is a lower bound on the number of walks and is likely the right asymptotics.
Jul 31, 2011 at 16:59 history answered Derrick Stolee CC BY-SA 3.0