Abstract
The theory of binomial posets developed in the previous chapter sheds considerable light on the “meaning” of generating functions and reduces certain types of enumerative problems to a routine computation. However, it does not seem worthwhile to attack more complicated problems from this point of view. The remainder of this book will for the most part be concerned with other techniques for obtaining and analyzing generating functions. We first consider the simplest general class of generating functions, namely, the rational generating functions. In this section we will concern ourselves with rational generating functions in one variable; that is, generating functions of the form \( F(x) = \sum\nolimits_{n \geqslant 0} {f(n)x^n } \) that are rational functions in the ring \( C[[x]] \). This means that there exist polynomials\( P(x),Q(x) \in {\text{C}}[x] \) such that \( F(x) = P(x)Q(x)^{ - 1} \) in \( C[[x]] \). Here it is assumed that \( Q(0) \ne 0 \), so that \( Q(x)^{ - 1} \) exists in \( C[[x]] \).
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© 1986 Wadsworth, Inc., Belmont, California 94002
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Stanley, R.P. (1986). Rational Generating Functions. In: Enumerative Combinatorics. The Wadsworth & Brooks/Cole Mathematics Series, vol 1. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-9763-6_4
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DOI: https://doi.org/10.1007/978-1-4615-9763-6_4
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