A famous conjecture of Stanley states that his chromatic symmetric function distinguishes trees. As a quasisymmetric analogue, we conjecture that the chromatic quasisymmetric function of Shareshian and Wachs and of Ellzey distinguishes directed trees. This latter conjecture would be implied by an affirmative answer to a question of Hasebe and Tsujie about the P-partition enumerator distinguishing posets whose Hasse diagrams are trees. They proved the case of rooted trees and our results include a generalization of their result.
Full text attached
Aval, Jean-Christophe; Djenabou, Karimatou; and McNamara, Peter R. W.. "Quasisymmetric Functions Distinguishing Trees." (2023) : 595-614.