Combinatorial statistics, probability and moment sequences

1. We present a generating function CF, 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.

   Under mild conditions on the parameters, the sequences arising from CF are moment sequences, and this continued fraction captures a variety of combinatorial sequences, counting various kinds of permutations, set partitions and perfect matchings, In particular, it characterizes the moment sequences associated to the numbers of permutations avoiding classical, vincular and consecutive patterns of length 3.

2. Generalizing the notion of descent set of a permutation, we study the number of permutations with a given, arbitrary consecutive pattern occurring at fixed positions. When these positions are at regular intervals we get an enumerating sequence for the permutations in question. We outline a recursive formula for all such 'regular' sequences, and conjecture that they are moment sequences.

The fourteen combinatorial statistics mentioned in 1. above naturally generalize to colored permutations, and, as an infinite family of statistics, to the k-arrangements: permutations with k-colored fixed points (where 0-arrangements are the derangements). The 2-arrangements were called simply arrangements by Comtet, and decorated permutations by Postnikov in his study of total positivity and Grassmannians, but for k>2 these do not seem to have been studied much, although they have many interesting properties, and many more yet to be discovered.

This is joint work with Natasha Blitvić and with Blitvić and Slim Kammoun, respectively.