Enumeration of unsensed orientable and non-orientable maps

Enumeration of 3-regular one-face maps on orientable or non-orientable surface up to all symmetries

“…In the recently published paper [16] Bernardi and Chapuy using the approach similar to [15] obtained exact formulas for counting cubic and precubic maps on non-orientable surfaces. These results along with the results of the work [11] allowed us to solve the problem of enumerating 3-regular one-face maps on both orientable and non-orientable surfaces up to all symmetries completely. To the best of our knowledge there are no published analytical results on enumerating such maps for arbitrary values of g.…”

“…Next, consider the case of unsensed maps on an orientable surface X + χ of Euler characteristic χ. In this case, for counting maps with n edges, instead of (1) we need to use the formula given in [11]:…”

“…Finally, consider the case of non-orientable surfaces. The number τX − χ (n) of unsensed maps on a non-orientable surface X − χ of Euler characteristic χ is calculated by the formula [11] τX…”

“…In the paper [10] analytical formulas for the numbers of unsensed r-regular maps on the torus were obtained. Finally, in [11] the problem of enumeration of unlabelled maps on genus g surfaces was solved in the most general formulation for the first time. The obtained general formulas express the numbers of such maps in the form of a linear combination of numbers of quotient maps on cyclic orbifolds with integer coefficients.…”

“…In [11] it was pointed out that with the help of these results one can enumerate different types of unsensed maps on surfaces of a given genus g (regular maps, one-face maps, etc.) assuming that we are able to enumerate quotient maps on orbifolds.…”