A Coloring Problem

Introduction. A well-known algorithm for coloring the vertices of a graph is the “greedy algorithm”: given a totally ordered set of colors, each vertex of the graph (taken in some order) is colored with the least color not already used to color an adjacent vertex. When applied to a path graph with at least two vertices, the algorithm uses either 2 or 3 colors, depending on the order in which the vertices are colored. I. Bouwer and Z. Star [1] solved the problem of counting the number of vertex orderings for a path of n vertices (out of n! possible vertex orderings) for which the greedy algorithm uses only two colors. They expressed the number of 2-color vertex orderings in terms of the number O(n) of 2-color vertex orderings in which the first vertex to be colored occurs in an odd position. They then found recurrences for O(2m) and O(2m + 1) that led to differential equations for the exponential generating functions