An algorithm for map enumeration

Abstract

Bauer and Itzykson showed that associated to each labeled map embedded on an oriented Riemann surface there was a group generated by a pair of permutations. From this result an algorithm may be constructed for enumerating labeled maps, and this construction is easily augmented to bin the numbers by the genus of the surface the map is embedded in. The results agree with the calculations of Harer and Zagier of 1-vertex maps; with those of Bessis, Itzykson, and Zuber of 4-valent maps; and with those of Ercolani, McLaughlin, and Pierce for $2ν$-valent maps. We then modify this algorithm to one which counts unoriented maps or Mobius graphs. The results in this case agree with the calculation of Goulden and Jackson on 1-vertex unoriented maps.