Melting lollipop chromatic quasisymmetric functions and Schur expansion of unicellular LLT polynomials

• Published in: Discrete mathematics Vol. 343; no. 3; p. 111728
• Main Authors: Huh, JiSun, Nam, Sun-Young, Yoo, Meesue
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.03.2020
• Subjects: [formula omitted]-free poset; [formula omitted]-positivity; Chromatic quasisymmetric function; LLT polynomials; Lollipop graph; Natural unit interval order; LLT polynomials; (3+1)-free poset; Lollipop graph; Natural unit interval order; Chromatic quasisymmetric function; e-positivity
• ISSN: 0012-365X
• DOI: 10.1016/j.disc.2019.111728

In this work, we generalize and utilize the linear relations of LLT polynomials introduced by Lee (2017). By using the fact that the chromatic quasisymmetric functions and the unicellular LLT polynomials are related via plethystic substitution and thus they satisfy the same linear relations, we can apply the linear relations to both sets of functions. As a result, in the chromatic quasisymmetric function side, we find a class of e-positive graphs, called melting lollipop graphs, and explicitly prove the e-unimodality. On the unicellular LLT side, we obtain Schur expansion formulas for LLT polynomials corresponding to certain set of graphs, namely, complete graphs, path graphs, lollipop graphs and melting lollipop graphs.