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Hi all:

Sorry for my bad wordy title, my question is:


Given a m*n 4-connected grid of points each of which is labelled with an integer $\in\{1,...,k\}$. We say a label configuration z is feasible (k-connected-components partition) if points of the same label are connected on the grid and there are k different labels in total.


How many different feasible configuration are there?

Thanks for your help


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Constraining m=n and k=2, this is

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