Patterns and Fractions

We find, in the form of a continued fraction, the generating function for the number of (132)avoiding permutations that have a given number of (123) patterns, and show how to extend this to permutations that have exactly one (132) pattern. We find some properties of the continued fraction, which is similar to, though more general than, those that were studied by Ramanujan. the electronic journal of combinatorics 6 (1999), #Rxx 2 A (132) pattern (resp. a (123) pattern) in a permutation π of |π| letters is a triple 1 ≤ i < j < k ≤ n of indices for which π(i) < π(k) < π(j) (resp. π(i) < π(j) < π(k)). Let fr(n) denote the number of permutations π of n letters that have no (132) patterns and exactly r (123) patterns. Our main result is the following. Theorem 1 The generating function for the {fr(n)} is