Counting permutations by the number of vertical edges in their bargraphs

In this paper, we consider a new statistic on the set S_n of permutations of length n which records the number of vertical edges within their bargraph representations. In-deed, we determine the distribution of this statistic on a class of multi-permutations having S_n as a subset. We compute an explicit formula for the total number of vertical edges within all members of this class and also for the number of interior vertices in the corresponding bargraphs. A generating function formula is determined and some closely related statistics are considered. Finally, we show that the total number of vertical edges in all the members of S_n equals the number of runs in the members of S_n+1 by constructing an explicit bijection which applies more generally to multi-permutations.