An exponential generating function (EGF) is a formal power series of the form $$E(x) = \\sum_{n=0}^{\\infty} a_n \\frac{x^n}{n!}$$, where the coefficients $a_n$ represent the number of ways to arrange or select objects. This tool is particularly useful in combinatorics for counting permutations and labeled structures, connecting closely with concepts such as enumeration techniques and algebraic structures in combinatorics. The EGF effectively transforms problems in counting into operations on power series, allowing for elegant solutions to various combinatorial problems, including those involving integer partitions and their properties.
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