We define a new family F̃w(X) of generating functions for w ∈ S̃n which are affine analogues of Stanley symmetric functions. We establish basic properties of these functions including symmetry, dominance and conjugation. We conjecture certain positivity properties in terms of a subfamily of symmetric functions called affine Schur functions. As applications, we show how affine Stanley symmetric fu...

Stanley has studied a symmetric function generalization XG of the chromatic polynomial of a graph G. The innocent-looking Stanley-Stembridge Poset Chain Conjecture states that the expansion of XG in terms of elementary symmetric functions has nonnegative coefficients if G is a clawfree incomparability graph. Here we give a combinatorial interpretation of these coefficients by combining Gasharov...