Tchebyshev Posets

We construct for each n an Eulerian partially ordered set Tn of rank n + 1 whose ce-index provides a non-commutative generalization of the n-th Tchebyshev polynomial. We show that the order complex of each Tn is shellable, homeomorphic to a sphere, and that its face numbers minimize the expression max|x|≤1 ∣∣∣∑nj=0(fj−1/fn−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.