Chromatic symmetric functions from the modular law

Abstract

In this article we show how to compute the chromatic quasisymmetric function of indifference graphs from the modular law introduced in [GP13]. We provide an algorithm which works for any function that satisfies this law, such as unicellular LLT polynomials. When the indifference graph has bipartite complement it reduces to a planar network, in this case, we prove that the coefficients of the chromatic quasisymmetric function in the elementary basis are positive unimodal polynomials and characterize them as certain q-hit numbers (up to a factor). Finally, we discuss the logarithmic concavity of the coefficients of the chromatic quasisymmetric function.

A symmetric function of increasing forests

“…The main idea to prove these results is to use the fact that csf q and LLT are completely determined by certain linear relations and their values at the complete graphs, as proved in [AN20]. Consequently, it is enough to prove that the right-hand sides of Equations ( 1e) and (1f) also satisfy these relations.…”