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

We prove that two horizontal-strip LLT polynomials are equal if the associated weighted graphs defined by the author in a previous paper are isomorphic. This provides a sufficient condition for equality of horizontal-strip LLT polynomials and yields a well-defined LLT polynomial indexed by a weighted graph. We use this to prove some new relations between LLT polynomials and we explore a connection with extended chromatic symmetric functions.

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

Keywords: Chromatic symmetric function, LLT polynomial, Hall-Littlewood polynomial, interval graph, Schur function, weighted graph

Lascoux, Leclerc, and Thibon defined a remarkable family of symmetric functions that are q-deformations of products of skew Schur functions. These LLT polynomials G_\lambda(x;q) can be indexed by a tuple \lambda of skew diagrams. When each skew diagram is a row, we define a weighted graph \Pi(\lambda). We show that a horizontal-strip LLT polynomial is determined by this weighted graph. When \Pi(\lambda) has no triangles, we establish a combinatorial Schur expansion of G_\lambda(x;q). We also explore a connection to extended chromatic symmetric functions.