# LLT polynomials, elementary symmetric functions and melting lollipops

## TL;DR

This paper proposes a positive combinatorial formula for unicellular LLT polynomials in the elementary basis, proves positivity for melting lollipops graphs, and explores related recurrence relations and implications.

## Contribution

It introduces a new explicit positive combinatorial formula for unicellular LLT polynomials and proves positivity for a specific class of graphs called melting lollipops.

## Key findings

- Conjectured a combinatorial formula for LLT polynomials in elementary basis.
- Proved positivity for melting lollipops graphs.
- Established recurrence relations generating all unicellular LLT polynomials.

## Abstract

We conjecture an explicit positive combinatorial formula for the expansion of unicellular LLT polynomials in the elementary symmetric basis. This is an analogue of the Shareshian-Wachs conjecture and previously studied by Panova and the author in 2018. We show that the conjecture for unicellular LLT polynomials implies a similar formula for vertical-strip LLT polynomials. We prove positivity in the elementary basis in for the class of graphs called `melting lollipops' previously considered by Huh, Nam and Yoo. This is done by proving a curious relationship between a generalization of charge and orientations of unit-interval graphs. We also provide short bijective proofs of Lee's three-term recurrences for unicellular LLT polynomials and we show that these recurrences are enough to generate all unicellular LLT polynomials associated with abelian area sequences.

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Source: https://tomesphere.com/paper/1903.03998