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How to get the all the lists of preimages of a surjective map is the image is given?

Firstly, I want to construct a map with the preimage {{5}, {4, 3}, {4, 2}, {3, 2, 1}, {2, 2, 1}, {2, 1}} and the image {1, 1, 2, 2, 3, 4} correspondingly. So as you can see it's a surjective map. Then …
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