Some applications of Rees products of posets to equivariant gamma-positivity

The Rees product of partially ordered sets was introduced by Bj\"orner and Welker. Using the theory of lexicographic shellability, Linusson, Shareshian and Wachs proved formulas, of significance in the theory of gamma-positivity, for the dimension of the homology of the Rees product of a graded poset $P$ with a certain $t$-analogue of the chain of the same length as $P$. Equivariant generalizations of these formulas are proven in this paper, when a group of automorphisms acts on $P$, and are applied to establish the Schur gamma-positivity of certain symmetric functions arising in algebraic and geometric combinatorics.Comment: Final version, with a section on type B Coxeter complexes added; to appear in Algebraic Combinatoric

A combinatorial proof that Schubert vs. Schur coefficients are nonnegative

We give a combinatorial proof that the product of a Schubert polynomial by a Schur polynomial is a nonnegative sum of Schubert polynomials. Our proof uses Assaf's theory of dual equivalence to show that a quasisymmetric function of Bergeron and Sottile is Schur-positive. By a geometric comparison theorem of Buch and Mihalcea, this implies the nonnegativity of Gromov-Witten invariants of the Grassmannian.Comment: 26 pages, several colored figure

Symmetric unimodal expansions of excedances in colored permutations

We consider several generalizations of the classical $\gamma$-positivity of Eulerian polynomials (and their derangement analogues) using generating functions and combinatorial theory of continued fractions. For the symmetric group, we prove an expansion formula for inversions and excedances as well as a similar expansion for derangements. We also prove the $\gamma$-positivity for Eulerian polynomials for derangements of type $B$. More general expansion formulae are also given for Eulerian polynomials for $r$-colored derangements. Our results answer and generalize several recent open problems in the literature.Comment: 27 pages, 10 figure