Signs in the -index of Eulerian partially ordered sets

Author:
Margaret M. Bayer

Journal:
Proc. Amer. Math. Soc. **129** (2001), 2219-2225

MSC (2000):
Primary 06A07

DOI:
https://doi.org/10.1090/S0002-9939-00-05831-7

Published electronically:
December 28, 2000

MathSciNet review:
1823903

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Abstract | References | Similar Articles | Additional Information

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 -index. The -index of a polytope has all positive entries. An important open problem is to give the broadest natural class of Eulerian posets having nonnegative -index. This paper completely determines which entries of the -index are nonnegative for all Eulerian posets. It also shows that there are no other lower or upper bounds on -coefficients (except for the coefficient of ).

**1.**Margaret M. Bayer and Louis J. Billera,*Generalized Dehn-Sommerville relations for polytopes, spheres and Eulerian partially ordered sets*, Invent. Math.**79**(1985), 143-157. MR**86f:52010b****2.**Margaret M. Bayer and Gábor Hetyei,*Flag vectors of Eulerian partially ordered sets*, European J. Combin. (to appear).**3.**-,*Generalizations of Eulerian partially ordered sets, flag numbers, and the Möbius function*, preprint 2000.**4.**Margaret M. Bayer and Andrew Klapper,*A new index for polytopes*, Discrete Comput. Geom.**6**(1991), 33-47. MR**91k:52024****5.**Louis J. Billera and Richard Ehrenborg,*Monotonicity of the -index for polytopes*, Math. Z.**233**(2000), 421-441.**6.**Louis J. Billera and Gábor Hetyei,*Linear inequalities for flags in graded partially ordered sets*, J. Combin. Theory Ser. A**89**(2000), 77-104. CMP**2000:07****7.**Louis J. Billera and Niandong Liu,*Noncommutative enumeration in graded posets*, J. Algebraic Combin.**12**(2000), 7-24.**8.**Richard Ehrenborg and Margaret Readdy,*Coproducts and the -index*, J. Algebraic Combin.**8**(1998), 273-299. MR**2000b:52009****9.**Gil Kalai,*A new basis of polytopes*, J. Combin. Theory Ser. A**49**(1988), 191-209. MR**90a:52012****10.**Isabella Novik,*Lower bounds for the -index of odd-dimensional simplicial manifolds*, European J. Combin.**21**(2000), 533-541.**11.**Mark Purtill,*André permutations, lexicographic shellability and the -index of a convex polytope*, Trans. Amer. Math. Soc.**338**(1993), 77-104. MR**93j:52017****12.**Richard P. Stanley,*Flag -vectors and the -index*, Math. Z.**216**(1994), 483-499. MR**96b:06006****13.**-,*Positivity problems and conjectures in algebraic combinatorics*, Mathematics: Frontiers and Perspectives, Amer. Math. Soc., Providence, RI, 2000, pp. 295-319. CMP**2000:13**

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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:
https://doi.org/10.1090/S0002-9939-00-05831-7

Keywords:
Eulerian poset,
flag vector,
$cd$-index

Received by editor(s):
October 22, 1999

Received by editor(s) in revised form:
December 21, 1999

Published electronically:
December 28, 2000

Communicated by:
John R. Stembridge

Article copyright:
© Copyright 2000
American Mathematical Society