enumeration of colorings

Enumeration of colorings involves counting the number of distinct ways to assign colors to elements such as the vertices of a graph or the squares of a grid, following specific rules—like no two adjacent vertices sharing the same color. This process helps understand the diversity of color arrangements under certain constraints. It uses combinatorial mathematics and symmetry considerations to avoid counting equivalent colorings multiple times, providing a structured way to analyze and quantify possible configurations in problems spanning graph theory, puzzles, and network design.