t-bargraphs and t-compositions

In this paper, we consider several statistics on t-compositions. We exhibit a bijection between the set of t-compositions and a set of bargraphs, introduced in this paper, named t-bargraphs. As our main statistic is per–the perimeter of a given t-bargraph, we obtain the corresponding generating function. We find both an explicit and an asymptotic formula for the total length of the perimeter over all t-bargraphs with n-cells.