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

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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.
Harper L. H. Harper, *Stirling behavior is asymptotically normal*, Ann. Math. Statist. **38** (1967), 410–414.
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/.
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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.