Counterexamples to the Neggers-Stanley conjecture
Author:
Petter Brändén
Journal:
Electron. Res. Announc. Amer. Math. Soc. 10 (2004), 155-158
MSC (2000):
Primary 06A07, 26C10
DOI:
https://doi.org/10.1090/S1079-6762-04-00140-4
Published electronically:
December 24, 2004
MathSciNet review:
2119757
Full-text PDF Free Access
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Abstract: The Neggers-Stanley conjecture asserts that the polynomial counting the linear extensions of a labeled finite partially ordered set by the number of descents has real zeros only. We provide counterexamples to this conjecture.
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Branden P. Brändén, Sign-graded posets, unimodality of $W$-polynomials and the Charney-Davis conjecture, Electr. J. Combin. 11(2), #9.
Brenti F. Brenti, Unimodal, log-concave and Pólya frequency sequences in combinatorics, Mem. Amer. Math. Soc. 413, Amer. Math. Soc., Providence, 1989.
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Neggers J. Neggers, Representations of finite partially ordered sets, J. Comb. Inf. Syst. Sci. 3 (1978), 113–133.
Rahman Q. I. Rahman and G. Schmeisser, Analytic theory of polynomials, The Clarendon Press, Oxford University Press, Oxford, 2002.
ReinerWelker V. Reiner and V. Welker, On the Charney-Davis and Neggers-Stanley conjectures, preprint; available at http://www.math.umn.edu/~reiner/.
Simion R. Simion, A multiindexed Sturm sequence of polynomials and unimodality of certain combinatorial sequences. J. Comb. Theory, Ser. A 36 (1984), 15–22.
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Wagner D. Wagner, Enumeration of functions from posets to chains, Eur. J. Comb. 13 (1992), 313–324.
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Additional Information
Petter Brändén
Affiliation:
Department of Mathematics, Chalmers University of Technology and Göteborg University, S-412 96 Göteborg, Sweden
MR Author ID:
721471
Email:
branden@math.chalmers.se
Keywords:
Neggers-Stanley conjecture,
partially ordered set,
linear extension,
real roots
Received by editor(s):
August 31, 2004
Published electronically:
December 24, 2004
Communicated by:
Sergey Fomin
Article copyright:
© Copyright 2004
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
