Enumerating branched orientable surface coverings over a non-orientable surface

The isomorphism classes of several types of graph coverings of a graph have been enumerated by many authors [M. Hofmeister, Graph covering projections arising from finite vector space over finite fields, Discrete Math. 143 (1995) 87-97; S. Hong, J.H. Kwak, J.Lee, Regular graph coverings whose covering transformation groups have the isomorphism extention property, Discrete Math. 148 (1996) 85-105; J.H. Kwak, J.H. Chun, J. Lee, Enumeration of regular graph coverings having finite abelian covering transformation groups, SIAM J. Discrete Math. 11 (1998) 273-285; J.H. Kwak, J. Lee, Isomorphism classes of graph bundles, Canad. J. Math. XLII (1990) 747-761; J.H. Kwak, J. Lee, Enumeration of connected graph coverings, J. Graph Theory 23 (1996) 105-109]. Recently, Kwak et al [Balanced regular coverings of a signed graph and regular branched orientable surface coverings over a non-orientable surface, Discrete Math. 275 (2004) 177-193] enumerated the isomorphism classes of balanced regular coverings of a signed graph, as a continuation of an enumeration work done by Archdeacon et al [Bipartite covering graphs, Discrete Math. 214 (2000) 51-63] the isomorphism classes of branched orientable regular surface coverings of a non-orientable surface having a finite abelian covering transformation group. In this paper, we enumerate the isomorphism classes of connected balanced (regular or irregular) coverings of a signed graph and those of unbranched orientable coverings of a non-orientable surface, as an answer of the question raised by Liskovets [Reductive enumeration under mutually orthogonal group actions, Acta-Appl. Math. 52 (1998) 91-120]. As a consequence of these two results, we also enumerate the isomorphism classes of branched orientable surface coverings of a non-orientable surface.