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This is a similar question to Can this be counted with stars and bars method?, where I'm trying to understand how Burnside's Lemma is used to compute orbits of of a matrix. I've been working through the example of how this applies to a cube rotation or the bead arrangements, but what I don't really get is how this applies to a matrix.

Given this specific example where I have a 2x2 matrix where each cell can have 2 values (black/white), I know there are 7 orbits from manually drawing them out.

In reading Burnside's Lemma, I think I have to get to the form:

1/n(...) = 7

but I'm not sure how to create the terms inside the parenthesis. I'm not really looking for heavy mathematical notation here for the general case (if possible), but I'm looking for more of an explanation of how the equations translate to actual numbers in this particular case.

More generally, when the matrix is MxN and not NxN, how do I compute the number of orbits?

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    what is the group action? – 2017-01-08
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    @JorgeFernándezHidalgo any row can be swapped with any other row. any column can be swapped with any other column – 2017-01-08
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    yes. ${}{}{}{}{}{}$ – 2017-01-08
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    @JorgeFernándezHidalgo can you show me how that figures into the overall formula? Thanks! – 2017-01-08
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    Did you figure this out in the end? – 2017-07-15
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    I guess this post alread answers this question https://math.stackexchange.com/questions/2056708/can-this-be-counted-with-stars-and-bars-method – 2017-07-15

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