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Enumerative Combinatorics by Sergey Fomin, Richard P. Stanley

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5

Trees and the Composition ofGenerating Functions

5.1 The Exponential Formula

If F(x) and G(x) are formal power series with G(0) = 0, then we have seen (after Proposition 1.1.9) that the composition F(G(x)) is a well-defined formal power series. In this chapter we will investigate the combinatorial ramifications of power series composition. In this section we will be concerned with the case where F(x) and G(x) are exponential generating functions, and especially the case F(x) = ex.

Let us first consider the combinatorial significance of the product F(x)G(x) of two exponential generating functions

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Throughout this chapter K denotes a field of characteristic ...

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