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Tchebyshev Posets

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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Correspondence to Gábor Hetyei.

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Hetyei, G. Tchebyshev Posets. Discrete Comput Geom 32, 493–520 (2004). https://doi.org/10.1007/s00454-004-1115-5

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Keywords

  • Computational Mathematic
  • Special Class
  • Simplicial Complex
  • Order Complex
  • Recursive Structure