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CyborgTosser (talk | contribs) fxd formula, explain series expansion of ln(1+u) |
CyborgTosser (talk | contribs) multiset |
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where the expansion
:<math>\ln(1 + u) = \sum_{k = 1}^{\infty} \frac{(-1)^{k-1}u^{k}}{k} </math> was used to go from line 4 to line 5.
===Multiset===
The ''multiset construction'', denoted <math>\mathcal{A} = \mathfrak{M}\{\mathcal{B}\}</math> is a generalization of the set construction. In the set construction, each element can occur zero or one times. In a multiset, each element can appear an arbitrary number of times. Therefore,
:<math>\mathfrak{M}\{\mathcal{B}\} = \prod_{\beta \in \mathcal{B}} \mathfrak{G}\{\beta\}</math>
This leads to the relation
:<math>\begin{matrix} A(z) & = & \prod_{\beta \in \mathcal{B}} (1 - z^{|\beta|})^{-1} \\
& = & \prod_{n = 1}^{\infty} (1 - z^{n})^{-B_{n}} \\
& = & \exp \left ( \ln \prod_{n = 1}^{\infty} (1 - z^{n})^{-B_{n}} \right ) \\
& = & \exp \left ( \sum_{n=1}^{\infty}-B_{n} \ln (1 - z^{n}) \right ) \\
& = & \exp \left ( \sum_{k=1}^{\infty} \frac{B(z^{k})}{k} \right )
\end{matrix}
</math>
where, similar to the above set construction, we expand <math>\ln (1 - z^{n})</math>, swap the sums, and substitute for the OGF of <math>\mathcal{B}</math>.
[[Category:combinatorics]]
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