Circular Digraph Walks, $k$-Balanced Strings, Lattice Paths and Chebychev Polynomials

We count the number of walks of length $n$ on a $k$-node circular digraph that cover all $k$ nodes in two ways. The first way illustrates the transfer-matrix method. The second involves counting various classes of height-restricted lattice paths. We observe that the results also count so-called $k$-balanced strings of length $n$, generalizing a 1996 Putnam problem.