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 In this talk we study a counting problem that originated on Mathematics Stack Exchange: How many ways can a rectangular grid be partitioned into a prescribed number of connected polyominoes when the pieces are colored, and any two pieces that share an edge must have different colors? We organize these numbers using bivariate generating functions, where one variable records the length of the grid and the other records the number of pieces. Using generating functions, we obtain explicit rational expressions in the first nontrivial cases of two and three rows. We then recast the model in graph-theoretic terms by replacing grids with Cartesian products of a fixed graph and a path, and by counting properly colored partitions into connected blocks. This leads to analogous generating functions on graphs, including closed forms for specific families (such as complete graphs), and to a computational framework for exploring further examples. This is joint work with Diego Villamizar (Xavier University of Louisiana