An algorithm for map enumeration

TL;DR

This paper presents an algorithm for enumerating labeled maps on Riemann surfaces, extending to unoriented maps, and confirms its results with existing calculations for various map types and genera.

Contribution

It introduces a permutation-based algorithm for counting labeled and unoriented maps, matching known enumeration results and extending to new cases.

Findings

Algorithm matches known counts for 1-vertex, 4-valent, and 2ν-valent maps.

Extends enumeration to unoriented maps and Möbius graphs.

Results agree with prior calculations by Harer, Zagier, Bessis, Itzykson, Zuber, Goulden, and Jackson.

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 $2\nu$-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.