Semi-perimeter and inner site-perimeter of k-ary words and bargraphs

Given a bargraph B, a border cell of B is a cell of B that shares at least one common edge with an outside cell of B. Clearly, the inner site-perimeter of B is the number of border cells of B. A tangent cell of B is a cell of B which is not a border cell of B and shares at least one vertex with an outside cell of B. In this paper, we study the generating function for the number of k-ary words, represented as bargraphs, according to the number of horizontal steps, up steps, border cells and tangent cells. This allows us to express some cases via Chebyshev polynomials of the second kind. Moreover, we find an explicit formula for the number of bargraphs according to the number of horizontal steps, up steps, and tangent cells/inner site-perimeter. We also derive asymptotic estimates for the mean number of tangent cells/inner site-perimeter.