Hafnian

In mathematics, the hafnian of an adjacency matrix of a graph is the number of perfect matchings in the graph. It was so named by Eduardo R. Caianiello "to mark the fruitful period of stay in Copenhagen (Hafnia in Latin)."[1]

The hafnian of a 2n × 2n symmetric matrix is computed as

${\displaystyle \operatorname {haf} (A)={\frac {1}{n!2^{n}}}\sum _{\sigma \in S_{2n}}\prod _{j=1}^{n}A_{\sigma (2j-1),\sigma (2j)},}$

where ${\displaystyle S_{2n}}$ is the symmetric group on [2n].[2]

Equivalently,

${\displaystyle \operatorname {haf} (A)=\sum _{M\in {\mathcal {M}}}\prod _{\scriptscriptstyle (u,v)\in M}A_{u,v}}$

where ${\displaystyle {\mathcal {M}}}$ is the set of all 1-factors (perfect matchings) on the complete graph ${\displaystyle K_{2n}}$, namely the set of all ${\displaystyle (2n)!/(n!2^{n})}$ ways to partition the set ${\displaystyle \{1,2,\dots ,2n\}}$ into ${\displaystyle n}$ subsets of size ${\displaystyle 2}$.[3][4]