Discrete & Computational Geometry
Enumerative combinatorics
Broken circuit complexes: factorization and generalizations
Journal of Combinatorial Theory Series B
Enumeration of functions from posets to chains
European Journal of Combinatorics
Permutation statistics of indexed permutations
European Journal of Combinatorics
q-Eulerian polynomials arising from Coxeter groups
European Journal of Combinatorics
Descents and one-dimensional characters for classical Weyl groups
Discrete Mathematics
Non-crossing partitions for classical reflection groups
Discrete Mathematics
On the Neggers-Stanley conjecture and the Eulerian polynomials
Journal of Combinatorial Theory Series A
Real Root Conjecture Fails for Five- and Higher-Dimensional Spheres
Discrete & Computational Geometry
The roots of the independence polynomial of a clawfree graph
Journal of Combinatorial Theory Series B
The poset of positive roots and its relatives
Journal of Algebraic Combinatorics: An International Journal
Special simplices and Gorenstein toric rings
Journal of Combinatorial Theory Series A
Actions on permutations and unimodality of descent polynomials
European Journal of Combinatorics
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For a graded naturally labelled poset P, it is shown that the P-Eulerian polynomial W(P, t): = Σw ∈ L(P) tdes(w) counting linear extensions of P by their number of descents has symmetric and unimodal coefficient sequence, verifying the motivating consequence of the Neggers-Stanley conjecture on real zeroes for W(P, t) in these cases. The result is deduced from McMullen's g-Theorem, by exhibiting a simplicial polytopal sphere whose h-polynomial is W(P, t).Whenever this simplicial sphere turns out to be flag, that is, its minimal non-faces all have cardinality two, it is shown that the Neggers-Stanley Conjecture would imply the Charney-Davis Conjecture for this sphere. In particular, it is shown that the sphere is flag whenever the poset P has width at most 2. In this case, the sphere is shown to have a stronger geometric property (local convexity), which then implies the Charney-Davis Conjecture in this case via a result from Leung and Reiner (Duke Math. J. 111 (2002) 253).It is speculated that the proper context in which to view both of these conjectures may be the theory of Koszul algebras, and some evidence is presented.