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

Abstract

We obtain explicit formulas for enumerating 3-regular one-face maps on orientable and nonorientable surfaces of a given genus g up to all symmetries. We use recent analytical results obtained by Bernardi and Chapuy for counting rooted precubic maps on non-orientable surfaces together with more widely known formulas for counting precubic maps on orientable surfaces. To take into account all symmetries we use a result of Krasko and Omelchenko that allows to reduce this problem to the problem of counting rooted quotient maps on orbifolds.