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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The toric $h$-vectors of partially ordered sets


Authors: Margaret M. Bayer and Richard Ehrenborg
Journal: Trans. Amer. Math. Soc. 352 (2000), 4515-4531
MSC (2000): Primary 06A07; Secondary 52B05
Published electronically: June 13, 2000
MathSciNet review: 1779486
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Abstract:

An explicit formula for the toric $h$-vector of an Eulerian poset in terms of the $\mathbf{cd}$-index is developed using coalgebra techniques. The same techniques produce a formula in terms of the flag $h$-vector. For this, another proof based on Fine's algorithm and lattice-path counts is given. As a consequence, it is shown that the Kalai relation on dual posets, $g_{n/2}(P)=g_{n/2}(P^*)$, is the only equation relating the $h$-vectors of posets and their duals. A result on the $h$-vectors of oriented matroids is given. A simple formula for the $\mathbf{cd}$-index in terms of the flag $h$-vector is derived.


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Additional Information

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

Richard Ehrenborg
Affiliation: School of Mathematics, Institute for Advanced Study, Olden Lane, Princeton, New Jersey 08540
Address at time of publication: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506
Email: jrge@ms.uky.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-00-02657-X
PII: S 0002-9947(00)02657-X
Keywords: Partially ordered set, $h$-vector, $\mathbf{c}\mathbf{d}$-index, coalgebra
Received by editor(s): February 15, 1998
Published electronically: June 13, 2000
Additional Notes: The first author was supported in part at MSRI by NSF grant #DMS 9022140
This work was begun when the second author was an H. C. Wang Assistant Professor at Cornell University and was completed at IAS under the partial support of NSF grant #DMS 97-29992 and NEC Research Institute, Inc.
Article copyright: © Copyright 2000 American Mathematical Society