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We use these to study the ratio of $|P(G_{pt,\\vec m},\\tau+1)|$ to the Tutte upper bound $(\\tau-1)^{n-5}$, where $\\tau=(1+\\sqrt{5} \\ )/2$ and $n$ is the number of vertices in $G_{pt,\\vec m}$. In particular, we calculate limiting values of this ratio as $n \\to \\infty$ for various families of planar triangulations. 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We use these to study the ratio of $|P(G_{pt,\\vec m},\\tau+1)|$ to the Tutte upper bound $(\\tau-1)^{n-5}$, where $\\tau=(1+\\sqrt{5} \\ )/2$ and $n$ is the number of vertices in $G_{pt,\\vec m}$. In particular, we calculate limiting values of this ratio as $n \\to \\infty$ for various families of planar triangulations. 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