The site-perimeter of words

We define [k] = {1, 2, 3, . . . , k} to be a (totally ordered) alphabet on k letters. A word w of length n on the alphabet [k] is an element of [k]^n. A word can be represented by a bargraph which is a family of column-convex polyominoes whose lower edge lies on the x-axis and in which the height of the i-th column in the bargraph equals the size of the i-th part of the word. Thus these bargraphs have heights which are less than or equal to k. We consider the site-perimeter, which is the number of nearest-neighbour cells outside the boundary of the polyomino. The generating function that counts the site-perimeter of words is obtained explicitly. From a functional equation we find the average site-perimeter of words of length n over the alphabet [k]. We also show how these statistics may be obtained using a direct counting method and obtain the minimum and maximum values of the site-perimeters.