OEIS A000305: Number of Certain Rooted Planar Maps

In this episode we explore OEIS A000305, which counts rooted, non-separable planar maps with N edges. A planar map is a connected graph embedded in the plane without edge crossings; rooted means we designate a directed edge to fix a reference, and non-separable means removing any single vertex leaves the map connected. The sequence starts 1, 4, 18, 89, 466, … and has the elegant closed form A_N = 2·(3N−3)! / (N!·(2N−1)!). This result comes from Tutte’s pioneering generating-function approach, with Brown extending the study to related map classes. This is a classic example of how a concrete combinatorial counting problem yields a neat exact formula, a staple of the OEIS bridge between objects and numbers.

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