We obtain explicit formulas for the coefficients of asymptotic expansions in the domain of large deviations for the distributions of the number of cycles ν_n in a random permutation of degree n, that is, for the probability P {ν_n = N} under the condition that n, N → ∞ in such a way that 1 < α₀ ≤ α = n / N ≤ α₁ < ∞, where α₀, α₁ are constants. These formulas express the coefficients in terms of cumulants of the random variable which has the distribution of the logarithmic series with specially chosen parameter. For the cumulants of the third and fourth orders we give the corresponding values. We discuss the accuracy of the obtained approximations. If n, N → ∞ so that 0 < γ₀ ≤ γ = N / ln n ≤ γ₁ < ∞, where γ₀, γ₁ are constants, we give asymptotic estimates of the probabilities P {ν_n = N}, P {ν_n ≤ N}, P {ν_n ≥ N} with the remainder terms of order O ((ln n)^-2) uniform in γ ∈ [γ₀,γ₁]. The corresponding estimate for the probability P {ν_n = N} refines the previously known results for the case N = β ln n + o (ln n), where β is a positive constant.