We show that symmetry, represented by a graph's automorphism group, can be used to greatly reduce the computational work for the substitution method. This allows application of the substitution method over larger regions of the problem lattices, resulting in tighter bounds on the percolation threshold p c . We demonstrate the symmetry reduction technique using bond percolation on the (3, 12 2 ) lattice, where we improve the bounds on p c from (0.738598, 0.744900) to (0.739399, 0.741757), a reduction of more than 62% in width, from 0.006302 to 0.002358.

We describe how non-crossing partitions arise in substitution method calculations. By using efficient algorithms for computing non-crossing partitions we are able to substantially reduce the computational effort, which enables us to compute improved bounds on the percolation thresholds for three percolation models. For the Kagomé bond model we improve bounds from 0.5182 p c 0.5335 to 0.522197 p c 0.526873, improving the range from 0.0153 to 0.004676. For the (3, 12 2 ) bond model we improve bounds from 0.7393 p c 0.7418 to 0.739773 p c 0.741125, improving the range from 0.0025 to 0.001352. We also improve the upper bound for the hexagonal site model, from 0.794717 to 0.743359.

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