A combinatorial formula for LLT cumulants of melting lollipops in terms of spanning trees

TL;DR

This paper presents a combinatorial formula expressing LLT cumulants of melting lollipops as positive combinations of LLT polynomials indexed by spanning trees, confirming their Schur-positivity and relating to parking functions.

Contribution

It introduces a new combinatorial formula for LLT cumulants of melting lollipops using spanning trees, establishing positivity and Schur-positivity.

Findings

Proves a positive combinatorial formula for LLT cumulants.

Confirms Schur-positivity of these cumulants.

Expresses the formula in terms of parking functions for complete graphs.

Abstract

We prove a combinatorial formula for LLT cumulants of melting lollipops as a positive combination of LLT polynomials indexed by spanning trees. The result gives an affirmative answer to a general positivity question for this class of unicellular LLT cumulants, and gives an independent proof of their Schur-positivity. In the special case of the complete graph, we also express the formula in terms of parking functions.