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.

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.