The paper is devoted to the problem of enumerating r-regular maps on two simplest non-orientable surfaces, the projective plane and the Klein bottle, up to all symmetries (so-called unsensed maps). We obtain general formulas that reduce the problem of counting such maps to the problem of enumerating rooted quotient maps on orbifolds. In addition, we solve the problem of explicitly describing all cyclic orbifolds for such surfaces. We also derive recurrence relations for rooted quotient maps on orbifolds that can be orientable or non-orientable surfaces with branch points and/or boundary components. These results enable us to obtain explicit formulas for the numbers of unsensed maps on the projective plane and the Klein bottle by the number of edges.

该论文致力于在两个最简单的不可定向表面上枚举r -正则映射问题，投影平面和克莱因瓶，直到所有对称（所谓的无感知映射）。我们获得了通用公式，将计算此类映射的问题简化为枚举 orbifolds 上的有根商映射的问题。此外，我们解决了明确描述此类表面的所有循环 orbifolds 的问题。我们还推导了轨道上的有根商映射的递归关系，这些映射可以是具有分支点和/或边界分量的可定向或不可定向表面。这些结果使我们能够通过边的数量获得投影平面和克莱因瓶上未感知映射数量的明确公式。