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CyborgTosser (talk | contribs) multiset |
CyborgTosser (talk | contribs) other elementary constructions |
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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>.
===Other elementary constructions===
Other important elementary constructions are the ''cycle construction'' (<math>\mathfrak{C}\{\mathcal{B}\}</math>), which are like sequences except that cyclic rotations are not considered distinct, ''pointing'' (<math>\Theta\mathcal{B}</math>), in which each member of <math>\mathcal{B}</math> is augmented by a neutral (zero size) pointer to one of its atoms, and ''substitution'' (<math>\mathcal{B} \circ \mathcal{C}</math>), in which each atom in a member of <math>\mathcal{B}</math> is replaced by a member of <math>\mathcal{C}</math>.
The derivations for these constructions are too complicated to show here. Here are the results:
{|
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|'''Construction'''
|'''Generating function'''
|-
|<math>\mathcal{A} = \mathfrak{C}\{\mathcal{B}\}</math>
|<math>A(z) = \sum_{k=1}^{\infty} \frac{\phi(k)}{k} \ln \frac{1}{1 - B(z^{k})}</math> (where <math>\phi(k)</math> is the [[Euler totient function]])
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|<math>\mathcal{A} = \Theta\mathcal{B}</math>
|<math>A(z) = z\frac{d}{dz}B(z)</math>
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|<math>\mathcal{A} = \mathcal{B} \circ \mathcal{C}</math>
|<math>A(z) = B(C(z))</math>
|}
[[Category:combinatorics]]
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