# A new approach to $e$-positivity for Stanley's chromatic functions

## TL;DR

This paper introduces a new combinatorial object to study the $e$-positivity of Stanley's chromatic functions, proving positivity for specific coefficients in the case of $(3+1)$-free posets.

## Contribution

It presents a novel combinatorial model using correct sequences of unit interval orders to establish $e$-positivity results for certain chromatic coefficients.

## Key findings

- Proves positivity of specific coefficients $c_{n-k,1^k}$, $c_{n-2,2}$, $c_{n-3,2,1}$, $c_{2^k,1^{n-2k}}$
- Introduces correct sequences of unit interval orders as a combinatorial tool
- Establishes positivity results for $(3+1)$-free posets

## Abstract

In this paper, we study positivity phenomena for the $e$-coefficients of Stanley's chromatic function of a graph. We introduce a new combinatorial object: the {\em correct} sequences of unit interval orders, and using these, in certain cases, we succeed to construct combinatorial models of the coefficients appearing in Stanley's conjecture. Our main result is the proof of positivity of the coefficients $c_{n-k,1^k}$, $c_{n-2,2}$, $c_{n-3,2,1}$ and $c_{2^k,1^{n-2k}}$ of the expansion of the chromatic symmetric function in terms of the basis of the elementary symmetric polynomials for the case of $(3+1)$-free posets.