Visibility in non-crossing and non-nesting partitions

Given a set partition π, whose canonical sequential form is represented geometrically as a bargraph, let (Formula presented.) denote the number of non-adjacent columns that are mutually visible to one another. In this paper, we enumerate partitions avoiding either 1212 or 1221 (the so-called non-crossing and non-nesting partitions, respectively) according to the vis parameter. In order to write recurrences satisfied by the distribution polynomials of vis on the two classes, we consider an auxiliary statistic in each case specific to the class in question and determine its joint distribution with vis. We compute formulas explicitly for the generating functions of the respective distributions as well as for the total value of vis on each class, making use of the kernel method to solve the functional equations that arise. Finally, we consider the restriction of vis to the set of Carlitz non-crossing partitions, i.e. those in which no two adjacent entries are equal, and compute its distribution on this class.