Visibility in pattern-restricted permutations

Given (Formula presented.), let (Formula presented.) denote the number of pairs of non-adjacent columns in the bargraph representation of π that are mutually visible to one other. In this paper, we enumerate permutations avoiding a single pattern of length three according to the vis statistic. Up to symmetry, we need only consider the cases of avoiding 123, 312 or 231, which are seen to yield distinct distributions. We compute explicit formulas for the generating functions of these distributions, making use of the kernel method in the 123 and 312 cases. Simple explicit formulas are found for the sum of the vis values taken over all members of a particular class as well as for the number of members of a class assuming a given value of vis in several specific instances.