Euler characteristic from coset cycle counts ($\chi =n(2-2g_0-r)+{\sum }_{i=1}^r{c}_i$)

An $n$-sheeted surface cover of an orbifold of underlying genus $g_0$ with $r$ cone points has the displayed ordinary Euler characteristic, where $c_i$ is the number of cycles of the $i$th cone monodromy permutation. The ramification deficit over that point is $n-c_i$, so the formula follows by counting lifted vertices, edges and faces, or by the Riemann-Hurwitz formula. If every cone cycle has its full prescribed length, this equals $n{\chi }_{\mathrm{orb}}$.