A combinatorial expansion of vertical-strip LLT polynomials in the basis of elementary symmetric functions

Abstract

We give a new characterization of the vertical-strip LLT polynomials ${\mathrm{LLT}}_P(x;q)$ as the unique family of symmetric functions that satisfy certain combinatorial relations. This characterization is then used to prove an explicit combinatorial expansion of vertical-strip LLT polynomials in terms of elementary symmetric functions. Such formulas were conjectured independently by A. Garsia et al. and the first named author, and are governed by the combinatorics of orientations of unit-interval graphs. The obtained expansion is manifestly positive if $q$ is replaced by $q+1$, thus recovering a recent result of M. D'Adderio. Our results are based on linear relations among LLT polynomials that arise in the work of D'Adderio, and of E. Carlsson and A. Mellit. To some extent these relations are given new bijective proofs using colorings of unit-interval graphs. As a bonus we obtain a new characterization of chromatic quasisymmetric functions of unit-interval graphs.

arXiv comment49 pages. This version has updated .bib, and some improvements in section 6

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Cited by in corpus (5)

• Chromatic symmetric functions of Dyck paths and q-rook theory
• LLT cumulants of unicellular Young diagrams, parking functions and Schur positivity
• Chromatic symmetric functions of Dyck paths and q-rook theory (extended abstract)
• A unipotent realization of the chromatic quasisymmetric function
• LLT cumulants and graph coloring