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

TL;DR

This paper explores counting walks on circular digraphs and their connections to k-balanced strings and lattice paths, using transfer-matrix methods and combinatorial analysis to derive new counting formulas.

Contribution

It introduces a novel approach linking circular digraph walks, lattice paths, and k-balanced strings, extending previous combinatorial problems.

Findings

Derived explicit formulas for walks covering all nodes

Established connections between digraph walks and lattice path enumeration

Generalized a classic Putnam problem to broader combinatorial contexts

Abstract

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.