Exact results for average cluster numbers in bond percolation on infinite-length lattice strips

We calculate exact analytic expressions for the average cluster numbers 〈k〉_{Λ_{s}} on infinite-length strips Λ_{s}, with various widths, of several different lattices, as functions of the bond occupation probability p. It is proved that these expressions are rational functions of p. As special cases of our results, we obtain exact values of 〈k〉_{Λ_{s}} and derivatives of 〈k〉_{Λ_{s}} with respect to p, evaluated at the critical percolation probabilities p_{c,Λ} for the corresponding infinite two-dimensional lattices Λ. We compare these exact results with an analytic finite-size correction formula and find excellent agreement. We also analyze how unphysical poles in 〈k〉_{Λ_{s}} determine the radii of convergence of series expansions for small p and for p near to unity. Our calculations are performed for infinite-length strips of the square, triangular, and honeycomb lattices with several types of transverse boundary conditions.