Tchebyshev Posets (English)

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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435
Guest Editors#8217 Preface
Bayer, Margaret M. / Lee, Carl W. / Sturmfels, Bernd | 2004
435
Guest Editors’ Preface
Bayer, Margaret M. / Lee, Carl W. / Sturmfels, Bernd | 2004
437
The Partial-Fractions Method for Counting Solutions to Integral Linear Systems
Beck, Matthias | 2004
447
Isoradial Bodies
Brandenberg, René / Dattasharma, Abhi / Gritzmann, Peter / Larman, David | 2004
459
Vertices of Gelfand--Tsetlin Polytopes
Loera, Jesús A. / McAllister, Tyrrell B. | 2004
471
A Complexity Bound on Faces of the Hull Complex
Develin, Mike | 2004
481
Enumerative Properties of Ferrers Graphs
Ehrenborg, Richard / Willigenburg, Stephanie | 2004
493
Tchebyshev Posets
Hetyei, Gábor | 2004
521
Mixed Fibre Polytopes
McMullen, Peter | 2004
533
Inequalities for the h-Vectors and Flag h-Vectors of Geometric Lattices
Nyman, Kathryn / Swartz, Ed | 2004
549
Convex Combinatorial Optimization
Onn, Shmuel / Rothblum, Uriel G. | 2004
567
Non-Crossing Frameworks with Non-Crossing Reciprocals
Orden, David / Rote, Günter / Santos, Francisco / Servatius, Brigitte / Servatius, Herman / Whiteley, Walter | 2004
601
The Et-Construction for Lattices, Spheres and Polytopes
Paffenholz, Andreas / Ziegler, Günter M. | 2004
623
Graphs, Syzygies, and Multivariate Splines
Rose, Lauren L. | 2004
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