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match operator name to flajolet/sedgewick text "analytic combinatorics" I.2 II.2 |
match operator name to flajolet/sedgewick text "analytic combinatorics" I.2 II.2 |
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Many combinatorial classes can be built using these elementary constructions. For example, the class of plane [[tree (graph)|tree]]s (that is, trees [[embedding|embedded]] in the plane, so that the order of the subtrees matters) is specified by the [[recursion|recursive]] relation
:<math>\mathcal{G} = \mathcal{Z} \times \
In other words, a tree is a root node of size 1 and a sequence of subtrees. This gives
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Another example (and a classic combinatorics problem) is [[integer partition]]s. First, define the class of positive integers <math>\mathcal{I}</math>, where the size of each integer is its value:
:<math>\mathcal{I} = \mathcal{Z} \times \
The OGF of <math>\mathcal{I}</math> is then
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Now, define the set of partitions <math>\mathcal{P}</math> as
:<math>\mathcal{P} = \
The OGF of <math>\mathcal{P}</math> is
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