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# Definition & Example

- A
**perfect matching**of the set $\{1,2,3,\ldots,2n\}$ is a partition into blocks of size 2.

the 3 Perfect matchings of size 4 | ||

[(1,2),(3,4)] |
[(1,3),(2,4)] |
[(1,4),(2,3)] |

- There are $(2n-1)!! = 1 \cdot 3 \cdot 5 \cdot \cdots \cdot (2n - 1)$ such perfect matchings, see OEIS:A001147.

# Additional information

- Perfect matchings can also be seen as fixed point free involutions on $\mathcal{S}$.
- Perfect matchings have a matrix known as the Weingarten matrix which are used to compute polynomial integrals over the orthogonal group $O_N$ [CM].
- Perfect matchings correspond to Kekulé structures in chemistry and give important information about the chemical structure of compounds. Applications include estimation of resonance energy, estimation of π-electron energy, and estimation of bond lengths.
- Hall's marriage theorem provides a characterization of bipartite graphs which have a perfect matching.
- Tutte's theorem provides a characterization for arbitrary graphs which have a perfect matching.

# References

- [CM] Benoit Collins and Sho Matsumoto, On some properties of orthogonal Weingarten functions, arXiv:0903.5143

# Sage examples

# Technical information for database usage

- A perfect matching is uniquely represented as a sorted list of increasing pairs.
- Perfect matchings are graded by the size.
- The database contains all perfect matchings of size at most 10.

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