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Considering the partitions of a set into nonempty subsets, we obtain an expression for the number of all partitions of a given type. The chromatic polynomial of a graph subdivision is generalized, considering two sets of colors, and a general explicit expression is obtained for this generalization. Using these results, we determine the generalized chromatic polynomial for the particular case of complete graph subdivision., UDC 519.17 1. Introduction One of the most important problems studied in graph theory is the coloring of the vertices of a graph. A proper vertex-coloring is color assignment to [...]