  In this paper we provide analytical counting rules for the ground states and
the quasiholes of fractional Chern insulators with an arbitrary Chern number.
We first construct pseudopotential Hamiltonians for fractional Chern
insulators. We achieve this by mapping the lattice problem to the lowest Landau
level of a multicomponent continuum quantum Hall system with specially
engineered boundary conditions. We then analyze the thin-torus limit of the
pseudopotential Hamiltonians, and extract counting rules (generalized Pauli
principles, or Haldane statistics) for the degeneracy of its zero modes in each
Bloch momentum sector.
