Some properties of subgroup counting function

Subgroups of index n in G = Cp1 ∗ Cp2 ∗ · · · ∗ Cpk are in 1-1 correspondance with subgroups of index n in F [x1, x2, . . . , xk], the free group on k generators, containing the normal subgroup generated by {x1 1 , x p2 2 , . . . , x pk k }. Such subgroups in turn are in 1-1 correspondance with connected directed graph on n vertices, with a basepoint, with edges coloured in k colours, such that each vertex has 1 incoming and 1 outgoing edge of each colour and the colour j cycles are either loops or pj cycles. This correspondance is given by the fundamental group of the graph w.r.t the basepoint. So for G = Cp1 ∗ Cp2 , subgroups of index n correspond to bi-coloured (say with colours red and blue) graphs, such that red cycles are either loops or p1 cycles and blue cycles are either loops or p2 cycles.