Kasteleyn's Formula

Theorem

The number of perfect covers of a chessboard of dimensions $m \times n$ is given by the formula:

$\ds \prod_{j \mathop = 1}^{\ceiling {\frac m 2} } \prod_{k \mathop = 1}^{\ceiling {\frac n 2} } \paren {4 \cos^2 \frac {\pi j} {m + 1} + 4 \cos^2 \frac {\pi k} {n + 1} }$

This article is complete as far as it goes, but it could do with expansion. In particular: Add an examples section. Examples are always fun You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Expand}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Proof

This theorem requires a proof. In particular: Import proof from Kasteleyns paper You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Source of Name

This entry was named for Pieter Willem Kasteleyn.

Sources

• 1961: P.W. Kasteleyn: The statistics of dimers on a lattice. I. The number of dimer arrangements on a quadratic lattice (Physica Vol. 27, no. 12: pp. 1209 – 1225)
• 2021: Jay Cummings: Proofs ... (previous) ... (next): Chapter $1$: $1.1$ Chessboard Problems