Given a set partition π, whose canonical sequential form is represented geometrically as a bargraph, let $\mathrm{v}\mathrm{i}\mathrm{s}\left(\pi \right)$ 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.

给定一个集合分区π，其规范的顺序形式用几何图形表示为条形图，让$\mathrm{v}\mathrm{一世}\mathrm{s}（\pi ）$表示彼此可见的不相邻列的数量。在本文中，我们列举了避免使用1212或1221（所谓的非交叉和非嵌套）的分区分别根据vis参数进行分区。为了在两个类上编写由vis分布多项式满足的递归，我们在每种情况下都考虑特定于所讨论类的辅助统计量，并确定其与vis的联合分布。我们使用核方法求解出现的函数方程，从而显式地计算各个分布的生成函数以及vis的总值的公式。最后，我们考虑将vis限制在Carlitz非交叉分区的集合上，即没有两个相邻条目相等的分区，并计算其在此类上的分布。