Catalan continued fractions and increasing subsequences in permutations Original Research Article

We call a Stieltjes continued fraction with monic monomial numerators a Catalan continued fraction. Let ek(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 eks. Moreover, there is an invertible linear transformation that translates between linear combinations of eks 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.