**EXTENSIVELY EDITED by David Speyer**. The previous version was very confused, but Steven Sivek and Graham, in the comments, figured out what was going on.

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G. C. Shephard, in his paper "[Twenty Problems on Convex Polyhedra: Part I][1]", associates to a three dimensional polyhedron the sequence $(p_3, p_4, p_5, p_6, \ldots)$, with $p_k$ being the number of facets that are $k$-gons. He poses the problem of characterizing all sequences of integers which arise in this way.


Are there any developments and references on this problem?


  [1]: http://www.jstor.org/pss/3612678