The perimeter of words

We define $\left[k\right]=\{1,2,\dots ,k\}$ to be a (totally ordered) alphabet on $k$ letters. A word $w$ of length $n$ on the alphabet $\left[k\right]$ is an element of ${\left[k\right]}^n$. A word can be represented by a bargraph (i.e., by a column-convex polyomino whose lower edges lie on the $x$-axis) in which the height of the $i$th column 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 perimeter, which is the number of edges on the boundary of the bargraph. By way of Cramer’s method and the kernel method, we obtain the generating function that counts the perimeter of words. Using these generating functions we find the average perimeter of words of length $n$ over the alphabet $\left[k\right]$. We also show how the mean and variance can be obtained using a direct counting method.