The Enumeration of Irreducible Combinatorial Objects

Abstract A unique factorization theory for labelled combinatorial objects is developed and applied to enumerate several families of objects, including certain families of set partitions, permutations, graphs, and collections of subintervals of [1, n ]. The theory involves a notion of irreducibility with respect to set partitions and the enumeration formulas that arise result from a generalization of the well-known “exponential formula.”

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