Abstract
We introduce a new method for the enumeration of self-avoiding
walks based on the lace expansion. We also introduce an
algorithmic improvement, called the two-step method, for
self-avoiding walk enumeration problems. We obtain significant
extensions of existing series on the cubic and hypercubic lattices
in all dimensions $d 3$: we enumerate $32$-step self-avoiding
polygons in $d=3$, $26$-step self-avoiding polygons in $d=4$,
$30$-step self-avoiding walks in $d=3$, and $24$-step
self-avoiding walks and polygons in all dimensions $d \geq
4$.
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