Stanley character polynomials

Stanley considered suitably normalized characters of the symmetric groups on Young diagrams having a special geometric form, namely multirectangular Young diagrams. He proved that the character is a polynomial in the lengths of the sides of the rectangles forming the Young diagram and he conjectured an explicit form of this polynomial. This Stanley character polynomial and this way of parametrizing the set of Young diagrams turned out to be a powerful tool for several problems of the dual combinatorics of the characters of the symmetric groups and asymptotic representation theory, in particular to Kerov polynomials.Comment: Dedicated to Richard P. Stanley on the occasion of his seventieth birthda

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