  In parametric sequence alignment, optimal alignments of two sequences are
computed as a function of the penalties for mismatches and spaces, producing
many different optimal alignments. Here we give a 3/(2^{7/3}\pi^{2/3})n^{2/3}
+O(n^{1/3} \log n) lower bound on the maximum number of distinct optimal
alignment summaries of length-n binary sequences. This shows that the upper
bound given by Gusfield et. al. is tight over all alphabets, thereby disproving
the "square root of n conjecture". Thus the maximum number of distinct optimal
alignment summaries (i.e. vertices of the alignment polytope) over all pairs of
length-n sequences is Theta(n^{2/3}).
