The length of the initial longest increasing sequence in a permutation

Let Sn be the set of all permutations {1, 2, …, n} represented in cycle notation. Define an, m to number π ∈ such that length initial longest increasing sequence (ILIS) is at most m. For fixed m, we find exponential generating function for and give an asymptotic formula when n → ∞. Moreover, show expected value total ILIS cycles over e ∑j=1 1/j.