  In an influential 1981 paper, Guibas and Odlyzko constructed a generating
function for the number of length n strings over a finite alphabet that avoid
all members of a given set of forbidden substrings. Here we extend this result
to the case in which the strings are weighted. This investigation was inspired
by the problem of counting compositions of an integer n that avoid all
compositions of a smaller integer m, a notion which arose from the
consideration of one-sided random walks.
