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{\bf Evangelos Georgiadis, David Callan and Qing-Hu Hou}
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{\bf Circular Digraph Walks, $k$-Balanced Strings, Lattice Paths and Chebychev Polynomials}
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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.



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