Mean and variance of the longest alternating subsequence in a random separable permutation

Abstract

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 $\frac{16-11\sqrt2}2n\approx0.2218\thinspace n$.