A two-regular graph is a regular graph for which all local degrees are 2. A two-regular graph consists of one or more (disconnected) cycles.
The numbers of two-regular graphs on , 2, ... nodes are 0, 0, 1, 1, 1, 2, 2, 3, 4, 5, ... (OEIS A008483), which are equivalent to the numbers of partitions of into parts . The first few such graphs are illustrated above.
This sequence has closed form
(1)
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where is the partition function P. It also has generating function given by
(2)
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(3)
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(4)
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where is a q-Pochhammer symbol.