Approximating the expected number of inversions given the number of breakpoints

We look at a problem with motivation from computational biology: Given the number of breakpoints in a permutation (representing a gene sequence), compute the expected number of inversions that have occurred. For this problem, we obtain an analytic approximation that is correct within a percent or two. For the inverse problem, computing the expected number of breakpoints after any number of inversions, we obtain an analytic approximation with an error of less than a hundredth of a breakpoint.