The (ordinary) generating function whose coeff. of z^n tells you the number of permutations of length n avoiding the set of patterns, {[1, 2, 3, 4], [1, 3, 2, 4]} equals 6 ----- \ ) g[i](1, 1, 1, z) / ----- i = 1 where the vector of formal power series of length, 6 in the variables, x[1], x[2], x[3], x[4], x[5], x[6], z is the unique vector that satisfies the following system of functional equations 3 2 3 g[1](x[1], x[2], x[3], z) = z x[1] x[2] x[3] 2 x[3] g[3](x[1], 1, x[2], x[3]) x[3] g[3](x[1], 1, x[2] x[3], 1) + ------------------------------- - -------------------------------- x[3] - 1 x[3] - 1 2 x[3] g[5](x[1], x[2], 1, x[3]) x[3] g[5](x[1], x[2] x[3], 1, 1) + ------------------------------- - -------------------------------- x[3] - 1 x[3] - 1 3 2 3 g[2](x[1], x[2], x[3], z) = z x[1] x[2] x[3] x[3] x[2] g[1](1, x[1], x[2] x[3], 1) + ------------------------------------- x[2] - 1 x[3] x[2] g[1](1, x[1] x[2], x[3], 1) - ------------------------------------- x[2] - 1 x[3] x[2] g[3](x[1], 1, x[2] x[3], 1) + ------------------------------------- x[2] - 1 x[3] x[2] g[3](x[1] x[2], 1, x[3], 1) - ------------------------------------- x[2] - 1 x[3] x[2] g[5](x[1], x[2] x[3], 1, 1) + ------------------------------------- x[2] - 1 x[3] x[2] g[5](x[1] x[2], x[3], 1, 1) - ------------------------------------- x[2] - 1 3 2 3 g[3](x[1], x[2], x[3], z) = z x[1] x[2] x[3] 2 x[3] g[4](x[1], 1, x[2], x[3]) x[3] g[4](x[1], 1, x[2] x[3], 1) + ------------------------------- - -------------------------------- x[3] - 1 x[3] - 1 2 x[3] g[6](x[1], x[2], 1, x[3]) x[3] g[6](x[1], x[2] x[3], 1, 1) + ------------------------------- - -------------------------------- x[3] - 1 x[3] - 1 3 2 3 g[4](x[1], x[2], x[3], z) = z x[1] x[2] x[3] x[3] x[2] x[1] g[1](1, x[1] x[2], x[3], 1) + ------------------------------------------ x[1] - 1 x[3] x[2] x[1] g[1](1, x[2], x[3], 1) - ------------------------------------- x[1] - 1 x[3] x[2] x[1] g[3](x[1] x[2], 1, x[3], 1) + ------------------------------------------ x[1] - 1 x[3] x[2] x[1] g[3](x[2], 1, x[3], 1) - ------------------------------------- x[1] - 1 x[3] x[2] x[1] g[5](x[1] x[2], x[3], 1, 1) + ------------------------------------------ x[1] - 1 x[3] x[2] x[1] g[5](x[2], x[3], 1, 1) - ------------------------------------- x[1] - 1 3 2 3 g[5](x[1], x[2], x[3], z) = z x[1] x[2] x[3] x[3] x[2] g[2](1, x[1], x[2] x[3], 1) + ------------------------------------- x[2] - 1 x[3] x[2] g[2](1, x[1] x[2], x[3], 1) - ------------------------------------- x[2] - 1 x[3] x[2] g[4](x[1], 1, x[2] x[3], 1) + ------------------------------------- x[2] - 1 x[3] x[2] g[4](x[1] x[2], 1, x[3], 1) - ------------------------------------- x[2] - 1 x[3] x[2] g[6](x[1], x[2] x[3], 1, 1) + ------------------------------------- x[2] - 1 x[3] x[2] g[6](x[1] x[2], x[3], 1, 1) - ------------------------------------- x[2] - 1 3 2 3 g[6](x[1], x[2], x[3], z) = z x[1] x[2] x[3] x[3] x[2] x[1] g[2](1, x[1] x[2], x[3], 1) + ------------------------------------------ x[1] - 1 x[3] x[2] x[1] g[2](1, x[2], x[3], 1) - ------------------------------------- x[1] - 1 x[3] x[2] x[1] g[4](x[1] x[2], 1, x[3], 1) + ------------------------------------------ x[1] - 1 x[3] x[2] x[1] g[4](x[2], 1, x[3], 1) - ------------------------------------- x[1] - 1 x[3] x[2] x[1] g[6](x[1] x[2], x[3], 1, 1) + ------------------------------------------ x[1] - 1 x[3] x[2] x[1] g[6](x[2], x[3], 1, 1) - ------------------------------------- x[1] - 1 This took, 0.825, seconds