New Invariants for Permutations, Orders and Graphs

We study the symmetric function and polynomial combinatorial invariants of Hopf algebras of permutations, posets and graphs. We investigate their properties and the relations among them. In particular, we show that the chromatic symmetric function and many other invariants have a property we call positively $h$-alternating. This property of positively $h$-alternating leads to Schur positivity and $e$-positivity when applying the operator $\nabla$ at $q=1$. We conclude by showing that the invariants we consider can be expressed as scheduling problems.