29 research outputs found
Sign-graded posets, unimodality of -polynomials and the Charney-Davis Conjecture
We generalize the notion of graded posets to what we call sign-graded
(labeled) posets. We prove that the -polynomial of a sign-graded poset is
symmetric and unimodal. This extends a recent result of Reiner and Welker who
proved it for graded posets by associating a simplicial polytopal sphere to
each graded poset . By proving that the -polynomials of sign-graded
posets has the right sign at -1, we are able to prove the Charney-Davis
Conjecture for these spheres (whenever they are flag).Comment: 14 page
Actions on permutations and unimodality of descent polynomials
We study a group action on permutations due to Foata and Strehl and use it to
prove that the descent generating polynomial of certain sets of permutations
has a nonnegative expansion in the basis ,
. This property implies symmetry and unimodality. We
prove that the action is invariant under stack-sorting which strengthens recent
unimodality results of B\'ona. We prove that the generalized permutation
patterns and are invariant under the action and use this to
prove unimodality properties for a -analog of the Eulerian numbers recently
studied by Corteel, Postnikov, Steingr\'{\i}msson and Williams.
We also extend the action to linear extensions of sign-graded posets to give
a new proof of the unimodality of the -Eulerian polynomials of
sign-graded posets and a combinatorial interpretations (in terms of
Stembridge's peak polynomials) of the corresponding coefficients when expanded
in the above basis.
Finally, we prove that the statistic defined as the number of vertices of
even height in the unordered decreasing tree of a permutation has the same
distribution as the number of descents on any set of permutations invariant
under the action. When restricted to the set of stack-sortable permutations we
recover a result of Kreweras.Comment: 19 pages, revised version to appear in Europ. J. Combi
Counterexamples to the Neggers-Stanley conjecture
The Neggers-Stanley conjecture (also known as the Poset conjecture) asserts
that the polynomial counting the linear extensions of a partially ordered set
on by their number of descents has real zeros only. We provide
counterexamples to this conjecture.Comment: 4 page
Examples and counterexamples in Ehrhart theory
This article provides a comprehensive exposition about inequalities that the
coefficients of Ehrhart polynomials and -polynomials satisfy under various
assumptions. We pay particular attention to the properties of Ehrhart
positivity as well as unimodality, log-concavity and real-rootedness for
-polynomials.
We survey inequalities that arise when the polytope has different normality
properties. We include statements previously unknown in the Ehrhart theory
setting, as well as some original contributions in this topic. We address
numerous variations of the conjecture asserting that IDP polytopes have a
unimodal -polynomial, and construct concrete examples that show that these
variations of the conjecture are false. Explicit emphasis is put on polytopes
arising within algebraic combinatorics.
Furthermore, we describe and construct polytopes having pathological
properties on their Ehrhart coefficients and roots, and we indicate for the
first time a connection between the notions of Ehrhart positivity and
-real-rootedness. We investigate the log-concavity of the sequence of
evaluations of an Ehrhart polynomial at the non-negative integers. We
conjecture that IDP polytopes have a log-concave Ehrhart series. Many
additional problems and challenges are proposed.Comment: Comments welcome