# New Invariants for Permutations, Orders and Graphs

## TL;DR

This paper introduces new invariants for permutations, orders, and graphs based on symmetric functions, revealing properties like positive h-alternating behavior that imply Schur and e-positivity, and connects these invariants to scheduling problems.

## Contribution

It defines and analyzes new combinatorial invariants for permutations, posets, and graphs, establishing their positive h-alternating property and expressing them through scheduling problems.

## Key findings

- Invariants exhibit positively h-alternating property.
- Application of operator ∇ at q=1 yields Schur and e-positivity.
- Invariants can be represented as scheduling problems.

## Abstract

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.