Pattern distributions in Dyck paths with a first return decomposition constrained by height

We provide generating functions for the popularity and the distribution of patterns of length at most three over the set of Dyck paths having a first return decomposition constrained by height.     

A new statistic on Dyck paths for counting 3-dimensional Catalan words

A 3-dimensional Catalan word is a word on three letters so that the subword on any two letters is a Dyck path. For a given Dyck path D, a recently defined statistic counts the number of Catalan words with the property that any subword on two letters is exactly D. In this paper, we enumerate Dyck paths with this statistic equal to certain values, including all primes. The formulas obtained are in terms of Motzkin numbers and Motzkin ballot numbers.     

Indecomposable permutations, hypermaps and labeled Dyck paths

Hypermaps were introduced as an algebraic tool for the representation of embeddings of graphs on an orientable surface. Recently a bijection was given between hypermaps and indecomposable permutations; this sheds new light on the subject by connecting a hypermap to a simpler object. In this paper, a bijection between indecomposable permutations and labeled Dyck paths is proposed, from which a few enumerative results concerning hypermaps and maps follow. We obtain for instance an inductive formula for the number of hypermaps with n darts, p vertices and q hyperedges; the latter is also the number of indecomposable permutations of Sn with p cycles and q left-to-right maxima. The distribution of these parameters among all permutations is also considered.     

Differential equations driven by rough paths with jumps

We develop the rough path counterpart of Itô stochastic integration and differential equations driven by general semimartingales. This significantly enlarges the classes of (Itô/forward) stochastic differential equations treatable with pathwise methods. A number of applications are discussed.