Definition. The Pfaffian records signed pairings [fdip-008I]

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Let \(A^T=-A\) be a \(2m\times 2m\) matrix over \(\mathbb R\) or \(\mathbb C\). A perfect matching of \(\{1,\ldots ,2m\}\) partitions it into pairs \((i_1,j_1),\ldots ,(i_m,j_m)\), written with \(i_r<j_r\) and \(i_1<\cdots <i_m\). Let \(\varepsilon (M)\) be the sign of the permutation with word \(i_1,j_1,\ldots ,i_m,j_m\). The Pfaffian is \[\operatorname {Pf}(A) =\sum _M\varepsilon (M)\prod _{r=1}^m A_{i_rj_r}.\] The empty Pfaffian is \(1\). This is the matching formula in [etingof2024mathematical, v2, Section 9.9, pp. 130-131].

Grouping matchings by the partner \(j\) of \(1\) gives \[\operatorname {Pf}(A)= \sum _{j=2}^{2m}(-1)^j A_{1j} \operatorname {Pf}(A_{\widehat 1\widehat j}),\] where the indicated two rows and columns are deleted, retaining the original order. Indeed, moving \(j\) immediately after \(1\) crosses \(j-2\) entries. In dimensions two and 4 this reads \[\operatorname {Pf}\begin {pmatrix}0&a\\-a&0\end {pmatrix}=a,\] \[\operatorname {Pf}(A)=A_{12}A_{34}-A_{13}A_{24}+A_{14}A_{23}.\] Each term pairs every index exactly once; unlike a determinant expansion, it sums over pairings rather than all permutations.