Counterexamples to the poset conjectures of Neggers, Stanley, and Stembridge

By John R. Stembridge. 2006

47 citations recorded by the source corpus.

We provide the first counterexamples to Neggers' 1978 conjecture and Stembridge's 1997 conjecture that the generating functions for descents and peaks in the linear extensions of naturally labeled posets should have all real zeros. We also provide minimum-sized counterexamples to a generalization of the Neggers conjecture due to Stanley that was recently disproved by Branden.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/5c8ac499a66116ccc1bfef7a2c22d456f1e057e2

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In Algebra and Number Theory

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Citations in the corpus

Unimodality, log-concavity, real-rootedness and beyond (2015, background): “ot; 1015 with 16 vertices (for exact counts, see the work of Brinkmann and McKay [BM]). 3 Figure 2. Counterexamples to Neggers conjecture (left) and the Neggers{Stanley conjecture (right), taken from [97]. Problem 1. Suppose all zeros of P n k=0 a kx k are nonpositive. If d2N, are all zeros of Xn k=”
Schur positivity and the q-log-convexity of the Narayana polynomials (2008, background): “It is also worth mentioning that the Schur expansion of the product of two Schur functions indexed by partitions with only two-columns turns out to be multiplicity-free if one is indexed by a rectangular shape; see Stembridge [31].”
Coxeter cones and their h-vectors (2008, methodology): “Motivated by a previous construction in the special case of posets (see Section 4 of [ St2 ]), we define this lattice equivalence to be a shift equivalence if translation by ν changes the height of each lattice point in �(�, λ) by some constant k. Recalling (4.2), this means”
On adding a list of numbers (and other one-dependent determinantal processes) (2009, result): “For background and first results, see Brenti [8] or Stembridge [72].”