Hafnian

Not to be confused with Hafnium.

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)."

The hafnian of a $2n\times 2n$ symmetric matrix is computed as

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

where $S_{2n}$ is the symmetric group on $[2n]=\{1,2,...,2n\}$.

Equivalently,

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

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

The permanent and the hafnian are related as $\operatorname {per} (A)=\operatorname {haf} {\begin{pmatrix}&A\\A^{T}&\\\end{pmatrix}}$.