Tchebyshev posets
Given a locally finite partially ordered set \(Q\), a second poset \(T(Q)\) is described. When \(Q\) is an Eulerian poset, often the same is true of \(T(Q)\). The main objects of study, the finite Eulerian posets \(T_n\) (for \(n = 1, 2, \ldots\)), are then obtained as intervals in the Eulerian poset \(T(P)\), where \(P\) is a certain relatively simple locally finite Eulerian poset. The \(cd\)-index and the \(ce\)-index of a finite Eulerian partially ordered set are each polynomials in two noncommuting variables (\(c, d\) or \(c, e\)) derived from the flag-\(f\)-vector of the poset. The flag \(f\)-vector of \(T_n\) is studied to the extent that the \(cd\)-index and the \(ce\)-index are found. The \(n\)-th Tchebyshev polynomial arises from the \(ce\)-index upon evaluating the \(ce\)-index at \(c=x\), \(e=1\). It is shown that the dual of \(T_n\) is the poset of faces of a CW-complex which is obtained from a subdivision of the boundary of a cross-polytope. An extremal property possessed by the \(f\)-vector of the simplicial complex of chains of \(T_n\) among \(f\)-vectors of simplicial complexes is described.
scientific article; zbMATH DE number 2138078