We construct for each n an Eulerian partially ordered set T n of rank n +1 whose ce-index provides a non-commutative generalization of the nth Tchebyshev polynomial. We show that the order complex of each T n is shellable, homeomorphic to a sphere, and that its face numbers minimize the expression max |x|≤1 | n j=0 ( f j−1 / f n−1 ) · 2 − j · (x − 1) j | among the f -vectors of all (n − 1)-dimensional simplicial complexes. The duals of the posets constructed have a recursive structure similar to face lattices of simplices or cubes, offering the study of a new special class of Eulerian partially ordered sets to test the validity of Stanley's conjecture on the non-negativity of the cd-index of all Gorenstein * posets.