Symbolic method (combinatorics): Difference between revisions

Revision as of 03:15, 3 January 2007

Revision as of 03:16, 3 January 2007

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This leads to the relation

:<math>\begin{matrixalign} 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{matrixalign}

</math>