Lattice path enumeration and Toeplitz matrices

This paper is about counting lattice paths. Examples are the paths counted by Catalan, Motzkin or Schröder numbers. We identify lattice paths with walks on some path-like graph. The entries of the nth power of the adjacency matrix are the number of paths of length nwith prescribed start and end position. The adjacency matrices turn out to be Toeplitz matrices. Explicit expressions for eigenvalues and eigenvectors of these matrices are known. This yields expressions for the numbers of paths in the form of trigonometric sums. We give many examples of known sequences that have such expressions. We also deal with cases where no explicit expressions for eigenvalues and eigenvectors of the relevant matrices are known. In some of these cases it is possible to use the characteristic polynomial to get linear recurrence relations for the numbers in question.