  Karlin and Altschul in their statistical analysis for multiple high-scoring
segments in molecular sequences introduced a distribution function which gives
the probability there are at least r distinct and consistently ordered segment
pairs all with score at least x. For long sequences this distribution can be
expressed in terms of the distribution of the length of the longest increasing
subsequence in a random permutation. Within the past few years, this last
quantity has been extensively studied in the mathematics literature. The
purpose of these notes is to summarize these new mathematical developments in a
form suitable for use in computational biology.
