  It is known that the monodromy group of each cover of a general curve of
genus g>3 equals either the symmetric or the alternating group. The classical
Catalan numbers count the minimal degree covers (with symmetric monodromy) of a
general curve of even genus. We solve the analogous problem for the alternating
group and we determine the number of alternating covers of minimal degree 2g+1
of a general curve of genus g.
