Permutations, moments, measures

Natasha Blitvić, Einar Steingrímsson

Published 2020-01-01Version 1

We present a continued fraction with 13 permutation statistics, several of them new, connecting a great number of combinatorial structures to a wide variety of moment sequences and their measures from classical and noncommutative probability. The Hankel determinants of these moment sequences are a product of $(p,q)$-factorials, unifying several instances from the literature. The corresponding measures capture as special cases several classical laws, such as the Gaussian, Poisson, and exponential, along with further specializations of the orthogonalizing measures in the Askey-Wilson scheme and several known noncommutative central limits. Statistics in our continued fraction generalize naturally to signed and colored permutations, and to the $k$-arrangements introduced here, permutations with $k$-colored fixed points.