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Signs in the $cd$-index of Eulerian partially ordered sets

Author(s): Margaret M. Bayer
Journal: Proc. Amer. Math. Soc. 129 (2001), 2219-2225.
MSC (2000): Primary 06A07
Posted: December 28, 2000
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Abstract:

A graded partially ordered set is Eulerian if every interval has the same number of elements of even rank and of odd rank. Face lattices of convex polytopes are Eulerian. For Eulerian partially ordered sets, the flag vector can be encoded efficiently in the $cd$-index. The $cd$-index of a polytope has all positive entries. An important open problem is to give the broadest natural class of Eulerian posets having nonnegative $cd$-index. This paper completely determines which entries of the $cd$-index are nonnegative for all Eulerian posets. It also shows that there are no other lower or upper bounds on $cd$-coefficients (except for the coefficient of $c^n$).

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Additional Information:

Margaret M. Bayer
Affiliation: Department of Mathematics, University of Kansas, Lawrence, Kansas 66045-2142
Email: bayer@math.ukans.edu

DOI: 10.1090/S0002-9939-00-05831-7
PII: S 0002-9939(00)05831-7
Keywords: Eulerian poset, flag vector, $cd$-index
Received by editor(s): October 22, 1999
Received by editor(s) in revised form: December 21, 1999
Posted: December 28, 2000
Communicated by: John R. Stembridge
Copyright of article: Copyright 2000, American Mathematical Society

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