  A permutation is \it separable \rm if it can be obtained from the singleton
permutation by iterating direct sums and skew sums. Equivalently, it is
separable if and only it avoids the patterns 2413 and 3142. Under the uniform
probability on separable permutations of $[n]$, let the random variable $A_n$
denote the length of the longest alternating subsequence. Also, let $A_n^{+,-}$
denote the length of the longest alternating subsequence that begins with an
ascent and ends with a descent, and define $A_n^{-,+}, A_n^{+,+}, A_n^{-,-}$
similarly. By symmetry, the first two and the last two of these latter four
random variables are equi-distributed. We prove that the expected value of any
of these five random variables behaves asymptotically as
$(2-\sqrt2)n\approx0.5858\thinspace n$. We also prove that the variance of any
of the four random variables $A_n^{\pm,\pm}$ behaves asymptotically as
$\frac{16-11\sqrt2}2n\approx0.2218\thinspace n$.
