Inclusion-exclusion relations on the triangular lattice

The combinatorial factors required for performing finite lattice series expansions on the triangular lattice are discussed. Series expansions can be obtained from a linear combination of free energies for finite convex hexagonal regions. An algorithm for calculating the combinatorial weights is given. A number of results are presented, showing that many cancellations occur and that not all convex hexagons of less than a given size are required. For a given degree of computational complexity (as given by the size of the largest transfer matrix) the expansion using hexagons of less than a certain perimeter and exploiting the full triangular symmetry gives 50% more series terms than the earlier approach which adds second-neighbour bonds to the square lattice and only exploits square lattice symmetries.