  We establish Guay-Paquet's unpublished linear relation between certain
chromatic symmetric functions by relating his algebra on paths to the
$q$-Klyachko algebra. The coefficients in this relations are $q$-hit
polynomials, and they come up naturally in our setup as connected remixed
Eulerian numbers, in contrast to the computational approach of
Colmenarejo-Morales-Panova. As Guay-Paquet's algebra is a down-up algebra, we
are able to harness algebraic results in the context of the latter and
establish results of a combinatorial flavour. In particular we resolve a
conjecture of Colmenarejo-Morales-Panova on chromatic symmetric functions. This
concerns the abelian case of the Stanley-Stembridge conjecture, which we
briefly survey.
