We construct for each nn an Eulerian partially ordered set T_nT_n of rank n+1n+1 whose cece-index provides a non-commutative generalization of the nnth Tchebyshev polynomial. We show that the order complex of each T_nT_n is shellable, homeomorphic to a sphere, and that its face numbers minimize the expression max_{|x| £ 1} |å_j=0ⁿ (f_j-1/f_n-1)·2^-j·(x-1)^j|\max_{|x|\leq 1} |\sum_{j=0}^n (f_{j-1}/f_{n-1})\cdot 2^{-j}\cdot (x-1)^j| among the ff-vectors of all (n-1)(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 Stanleys conjecture on the non-negativity of the cdcd-index of all Gorenstein^*^* posets.