Tchebyshev Posets (Englisch)

In: Discrete & Computational Geometry   ;  32 ,  4  ;  493-520  ;  2004

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Abstract 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 $n$th 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 $\max_{|x|\leq 1} |\sum_{j=0}^n (f_{j-1}/f_{n-1})\cdot 2^{-j}\cdot (x-1)^j|$ 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.

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