We consider three distinct topics of independent interest;
one in enumerative combinatorics, one in symmetric function theory, and
one in algebraic geometry. The topic in enumerative combinatorics
concerns a q-analog of a generalization of the Eulerian numbers, the one
in symmetric function theory deals with a refinement of Stanley's
chromatic symmetric functions, and the one in algebraic geometry deals
with a representation of the symmetric group on the cohomology of the
regular semisimple Hessenberg variety of type A. Our purpose is to
explore some connections between these topics and consequences of these
connections. This talk is based on joint work with John Shareshian.