We conjecture an explicit positive combinatorial formula for the expansion of unicellular LLT polynomials in the elementary symmetric basis. This is an analogue of the Shareshian–Wachs conjecture previously studied by Panova and the author in 2018. We show that the conjecture for unicellular LLT polynomials implies a similar formula for vertical-strip LLT polynomials. We prove positivity in the elementary symmetric basis for the class of graphs called “melting lollipops” previously considered by Huh, Nam and Yoo. This is done by proving a curious relationship between a generalization of charge and orientations of unit-interval graphs. We also provide short bijective proofs of Lee’s three-term recurrences for unicellular LLT polynomials, and we show that these recurrences are enough to generate all unicellular LLT polynomials associated with abelian area sequences.

我们猜想一个显式正组合公式，用于在基本对称基础上扩展单细胞LLT多项式。这与Panova及其作者先前在2018年研究的Shareshian-Wachs猜想类似。我们证明，单细胞LLT多项式的猜想暗含垂直条形LLT多项式的相似公式。我们证明了先前由Huh，Nam和Yoo所考虑的称为“融化棒棒糖”的一类图的基本对称性为正。这是通过证明电荷的泛化和单位间隔图的方向之间的奇怪关系来完成的。我们还提供了单细胞LLT多项式Lee的三项递归的短双射证明，