Enumeration of simple random walks and tridiagonal matrices

Abstract

We present some old and new results in the enumeration of random walks in one dimension, mostly developed in works of enumerative combinatorics. The relation between the trace of the $n$-th power of a tridiagonal matrix and the enumeration of weighted paths of $n$ steps allows an easier combinatorial enumeration of the paths. It also seems promising for the theory of tridiagonal random matrices .

several ref.and comments added, misprints corrected

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• Combinatorics of generalized Dyck and Motzkin paths
• A Novel Fibonacci Pattern in Pascal's Triangle
• Real symmetric random matrices and paths counting
• Factorization of the characteristic function of a Jacobi matrix