Poset generating functions and log-supermodularity

In joint work with T.Lam and A.Postnikov, we establish Schur positivity conjectures due to Okounkov and Fomin-Fulton-Li-Poon, and a special case of a conjecture by Lascoux-Leclerc-Thibon. We study log-supermodularity properties of more general poset generating functions, proving a positivity result in terms of fundamental quasisymmetric functions. We introduce quasisymmetric functions called wave Schur functions, and provide a Jacobi-Trudi like formula for them. Finally, we look at the related partial order on pairs of partitions introduced by J. Stembridge, and characterise maximal elements of that poset.