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How many monotone mappings are there between $P(E)$ (set of all subsets of $E$) and $P(F)$ if $\text{card}(E) = n$ and $\text{card}(F) = m$?

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    $\begingroup$ This is the number $f(m,n) $ of order ideals of the product $C_m\times C_2^n$, where $C_i$ is an $i$-element chain. The number $g(n)$ of order ideals of $C_2^n$ is "Dedekind's problems," oeis.org/A000372. It is unlikely that there is a simple formula or even a fast way to compute $g(n)$, so probably also $f(m,n)$. The techniques used to estimate $g(n)$ can be applied to $f(m,n)$. $\endgroup$ Commented Feb 22, 2014 at 16:25

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