For each integer partition µ, let F_µ(q,t) be the coefficient of x₁⋯x_n in the modified Macdonald polynomial H_µ. The polynomial F_µ(q,t) can be regarded as the Hilbert series of a certain doubly-graded S_n-module M_µ, or as a q,t-analogue of n! based on permutation statistics inv_µ and maj_µ that generalize the classical inversion and major index statistics. This paper uses the combinatorial definition of F_µ to prove some recursions characterizing these polynomials, and other related ones, when µ is a two-column shape. Our result provides a complement to recent work of Garsia and Haglund, who proved a different recursion for two-column shapes by representation-theoretical methods. For all µ, we show that F_µ(q,t) is divisible by certain q-factorials and t-factorials depending on µ. We use our recursion and related tools to explain some of these factors bijectively. Finally, we present fermionic formulas that express F_(2ⁿ)(q,t) as a sum of q,t-analogues of n!2ⁿ indexed by perfect matchings.