Which combinatorial sequences correspond to moments of probability measures on the real line? We present a generating function, as a continued fraction, for a 14-parameter family of integer sequences and interpret these in terms of statistics on permutations and other combinatorial objects. Special cases include several classical and noncommutative probability laws, and a substantial subset of the orthogonalizing measures in the q-Askey scheme of orthogonal polynomials.
This continued fraction captures a variety of combinatorial sequences. In particular, it characterizes the moment sequences associated to the numbers of permutations avoiding (classical and vincular) patterns of length three. This connection between pattern avoidance and classical and noncommutative probability is among several consequences that generalize and unify previous results in the literature.
The fourteen combinatorial statistics further generalize to colored permutations, and, as an infinite family of statistics, to the k-arrangements: permutations with k-colored fixed points, introduced here. This is joint work with Natasha Blitvić, Lancaster University.