Geodesic
Circular digraph walks, \(k\)-balanced strings, lattice paths and Chebychev polynomials
The electronic journal of combinatorics, Tome 15 (2008)

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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.
DOI : 10.37236/832
Classification : 05A05, 05A15, 33C45, 05C20, 05C38
Mots-clés : \(k\)-node circular digraph, counting walks of given length, transfer matrix method, generating function, Cramer's rule, Chebyshev polynomial, bad walks, height restricted lattice paths, \(k\)-balanced strings 
Evangelos Georgiadis; David Callan; Qing-Hu Hou. Circular digraph walks, \(k\)-balanced strings, lattice paths and Chebychev polynomials. The electronic journal of combinatorics, Tome 15 (2008). doi: 10.37236/832
@article{10_37236_832,
     author = {Evangelos Georgiadis and David Callan and Qing-Hu Hou},
     title = {Circular digraph walks, \(k\)-balanced strings, lattice paths and {Chebychev} polynomials},
     journal = {The electronic journal of combinatorics},
     year = {2008},
     volume = {15},
     doi = {10.37236/832},
     zbl = {1180.05002},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/832/}
}
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