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I don't quite know how to count the number of $k$-dimensional faces of a $4$-dimensional cyclic polytope (http://en.wikipedia.org/wiki/Cyclic_polytope) without using the standard formula.

Any advice?

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The Wikipedia page you linked gives an explicit formula for the faces of dimension 0 and 1, and you know there is just one face of dimension -1; the nullitope, and one face of dimension 4; the polytope itself. So, you just need to know the counts of the faces of dimension 2 and 3. The Dehn-Sommerville formula linked from the cyclic polytope page (upon setting k = -1, 0, 1 & 2) provides 4 equations, which suffice to solve for your two unknowns.

Also, there's an explicit formula here.

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