Sign-graded posets, unimodality of WW-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 $W$-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 $P$. By proving that the $W$-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

On positivity of Ehrhart polynomials

Ehrhart discovered that the function that counts the number of lattice points in dilations of an integral polytope is a polynomial. We call the coefficients of this polynomial Ehrhart coefficients, and say a polytope is Ehrhart positive if all Ehrhart coefficients are positive (which is not true for all integral polytopes). The main purpose of this article is to survey interesting families of polytopes that are known to be Ehrhart positive and discuss the reasons from which their Ehrhart positivity follows. We also include examples of polytopes that have negative Ehrhart coefficients and polytopes that are conjectured to be Ehrhart positive, as well as pose a few relevant questions.Comment: 40 pages, 7 figures. To appear in in Recent Trends in Algebraic Combinatorics, a volume of the Association for Women in Mathematics Series, Springer International Publishin

Stable polynomials, Lorentzian polynomials and discrete convexity

A multivariate polynomial is stable if it is nonzero whenever all its variables have positive imaginary parts. Lorentzian polynomials generalize homogeneous and stable polynomials. We discuss the spaces of stable and Lorentzian polynomials, their tropicalizations, and the connection to discrete convexity. This is based on joint work with June Huh.Non UBCUnreviewedAuthor affiliation: KTH Royal Institute of TechnologyResearche

Catalan Continued Fractions And Increasing Subsequences in Permutations

We call a Stieltjes continued fraction with monic monomial numerators a Catalan continued fraction. Let e k () be the number of increasing subsequences of length k + 1 in the permutation . We prove that any Catalan continued fraction is the multivariate generating function of a family of statistics on the 132-avoiding permutations, each consisting of a (possibly in- nite) linear combination of the e k s. Moreover, there is an invertible linear transformation that translates between linear combinations of e k s and the corresponding continued fractions