# Lollipop and lariat symmetric functions

## TL;DR

This paper derives an explicit e-positive formula for the chromatic symmetric function of lollipop graphs, shows the existence of infinitely many e-positive bases, and resolves six conjectures related to lariat graphs.

## Contribution

It provides a new explicit formula for lollipop graphs' chromatic symmetric functions and proves the existence of infinitely many e-positive bases in the symmetric functions algebra.

## Key findings

- Explicit e-positive formula for lollipop graphs
- Existence of infinitely many e-positive, Schur-positive bases
- Resolution of six conjectures on lariat graphs

## Abstract

We compute an explicit $e$-positive formula for the chromatic symmetric function of a lollipop graph, $L_{m,n}$. From here we deduce that there exist countably infinite distinct $e$-positive, and hence Schur-positive, bases of the algebra of symmetric functions whose generators are chromatic symmetric functions. Finally, we resolve 6 conjectures on the chromatic symmetric function of a lariat graph, $L_{n+3}$.