  The length is(w) of the longest increasing subsequence of a permutation w in
the symmetric group S_n has been the object of much investigation. We develop
comparable results for the length as(w) of the longest alternating subsequence
of w, where a sequence a,b,c,d,... is alternating if a>b<c>d<.... For instance,
the expected value (mean) of as(w) for w in S_n is exactly (4n+1)/6 if n>1.
