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  "body": "\n## Summary\n\nThe Pfaffian is a polynomial in the entries of a [[skew-symmetric matrix|0PBY4E9GQF]] of even size whose square is the [[determinant|00KDZRW4C0]].\nIt is a sum over the [[perfect matchings|0NHK9MFFW9]] of the index set, with signs.\nIt appears in the study of alternating forms and symplectic groups, and in the counting of perfect matchings of planar graphs.\n\n## Explanations\n\n### L1 human\n\nThe determinant of a skew-symmetric matrix of even size is always a perfect square of a polynomial in its entries.\nThe Pfaffian is that square root, with a fixed choice of sign.\nEach term of the Pfaffian pairs up the indices $1, \\dots, 2n$ and multiplies the matching entries.\nFor $A = \\begin{pmatrix} 0 \u0026 a \\\\ -a \u0026 0 \\end{pmatrix}$, the Pfaffian is $a$, and $\\det A = a^2$.\n",
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