Lower bounds for optimal alignments of binary sequences

In parametric sequence alignment, optimal alignments of two sequences are computed as a function of matches, mismatches and spaces, producing many different optimal alignments. In the two-parameter case, Gusfield et al shows that the number of distinct optimal alignment summaries for a pair of sequences is image. Here we construct binary sequences of length image with image distinct optimal alignment summaries. This shows that the upper bound given by Gusfield et al. is tight over all alphabets, thereby disproving the “image conjecture”. Thus the maximum number of distinct optimal alignment summaries over all pairs of length image sequences is image.