  We count permutations avoiding a nonconsecutive instance of a two- or
three-letter pattern, that is, the pattern may occur but only as consecutive
entries in the permutation. Two-letter patterns give rise to the Fibonacci
numbers. The counting sequences for the two representative three-letter
patterns, 321 and 132, have respective generating functions
(1+x^2)(C(x)-1)/(1+x+x^2-x C(x)) and C(x+x^3) where C(x) is the generating
function for the Catalan numbers.
