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Given a positive integer $k\ge2$, let be $f_k(m,n)$ the number of ways to cover an $m × n$ rectangle with $mn/k$ tiles ( $1×k$ or $k×1$)

$f_2(m,n)$ is kasteleyn formula

$f_k(m,n)$?

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    $\begingroup$ I'm sure there's no known exact formula (analogous to Fischer-Temperley/Kasteleyn) for this. $\endgroup$ Jul 25, 2022 at 13:13
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    $\begingroup$ Although, if say $m$ (and $k$) are fixed, then by standard techniques $f_k(m,n)$ as a function of $n$ satisfies a linear recurrence with constant coefficients, so its generating function is rational, etc. $\endgroup$ Jul 25, 2022 at 13:16

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