Harmonious coloring is defined as a (proper) vertex coloring in which every pair of colors appears on at most one pair of adjacent vertices. So given a path that contains $5$ vertices, we have assigned chromatic polynomial or ways to color a vertex using the definition of the harmonious coloring. So this is what we have come up to:
Is my way of assigning the ways to color the vertex is correct? Do I achieve a correct chromatic polynomial for a path using the definition of harmonious coloring?
Next, we have is a path with 11 vertices, the same application of the harmonious definition. Consider that this is a path where the assigned colors are the vertices.
red -- blue -- green -- black -- white -- red -- black -- blue -- white -- green -- red
what chromatic poly we can achieve in here?