Combinatorial Theory

In recent years, Alexandersson and others proved combinatorial formulas for the Schur function expansion of the horizontal-strip LLT polynomial \(G_{\boldsymbol\lambda}(\boldsymbol x;q)\) in some special cases. We associate a weighted graph \(\Pi\) to \(\boldsymbol\lambda\) and we use it to express a linear relation among LLT polynomials. We apply this relation to prove an explicit combinatorial Schur-positive expansion of \(G_{\boldsymbol\lambda}(\boldsymbol x;q)\) whenever \(\Pi\) is triangle-free. We also prove that the largest power of \(q\) in the LLT polynomial is the total edge weight of our graph.

Keywords: Charge, chromatic symmetric function, cocharge, Hall--Littlewood polynomial, jeu de taquin, LLT polynomial, interval graph, Schur function, Schur-positive, symmetric function.

Mathematics Subject Classifications: 05E05, 05E10, 05C15