P-partition products and fundamental quasi-symmetric function positivity
  We show that certain differences of products of $P$-partition generating functions are positive in the basis of fundamental quasi-symmetric functions L_\alpha. This result interpolates between recent Schur positivity and monomial positivity results of the same flavor. We study the case of chains in detail, introducing certain ``cell transfer'' operations on compositions and an interesting related ``L-positivity'' poset. We introduce and study quasi-symmetric functions called ``wave Schur functions'' and use them to establish, in the case of chains, that the difference of products we study is itself equal to a single generating function K_{P,\theta} for a labeled poset (P,\theta). In the course of our investigations we establish some factorization properties of the ring of quasisymmetric functions. 
Thomas Lam and Pavlo Pylyavskyy