A combinatorial Schur expansion of triangle-free horizontal-strip LLT polynomials

In recent years, Alexandersson and others proved combinatorial formulas for the Schur function expansion of horizontal-strip LLT polynomial \(G_{\boldsymbol\lambda}(\boldsymbol x;q)\) in some special cases. We associate a weighted graph \(\Pi\) to \(\boldsymbol\lambda\) we use it express linear relation among polynomials. apply this prove an explicit Schur-positive whenever is triangle-free. also that largest power \(q\) total edge weight our graph.Keywords: Charge, chromatic symmetric function, cocharge, Hall--Littlewood polynomial, jeu de taquin, interval graph, Schur-positive, function.Mathematics Subject Classifications: 05E05, 05E10, 05C15