Motivation: I am working on a problem that reduces to finding simplicial complexes given some data (details are unnecessary), but all I have managed to wrangle from my input is the number of cells of each dimension. What I want is to find the possible complexes with these numbers of cells. This really is not my field of expertise (if anything is), so I turn to your collective wisdom.

So suppose I am given a vector of positive integers, e.g. $(8,12,6,1)$. What simplicial complexes can I build with number of $k$-cells given by the vector? (i.e. in the example, 8 0-cells, 12 1-cells, etc.)

I expect that for most such vectors there would be several possible complexes one could build out of them, but I have no idea if (a) this is true, or (b) how to find them, or even (c) to find the number of possibilities. Has anyone done anything similar before? Is anything known about this? I'd appreciate any suggestions or references.

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(I was not really sure which tags to apply. Feel free to retag.)