Catalan continued fractions and increasing subsequences in permutations

Abstract

We call a Stieltjes continued fraction with monic monomial numerators a Catalan continued fraction. Let e k ( π) be the number of increasing subsequences of length k+1 in the permutation π. We prove that any Catalan continued fraction is the multivariate generating function of a family of statistics on the 132-avoiding permutations, each consisting of a (possibly infinite) linear combination of the e k s. Moreover, there is an invertible linear transformation that translates between linear combinations of e k s and the corresponding continued fractions. Some applications are given, one of which relates fountains of coins to 132-avoiding permutations according to number of inversions. Another relates ballot numbers to such permutations according to number of right-to-left maxima.