LLT cumulants of unicellular Young diagrams, parking functions and Schur positivity

TL;DR

This paper presents a combinatorial formula for LLT cumulants of unicellular shapes, linking parking functions and Cayley trees, and proves Schur positivity conjectures.

Contribution

It introduces a new combinatorial formula for LLT cumulants of unicellular shapes, confirming Schur positivity conjectures.

Findings

Formula for LLT cumulants in terms of parking functions and Cayley trees

Proof of Schur positivity for these cumulants

Connections between combinatorial structures and algebraic positivity

Abstract

We give a combinatorial formula for LLT cumulants of unicellular unilevelled shapes in terms of parking functions and Cayley trees. Our formula implies previously conjectured Schur positivity.